Summary of results: The conclusions are that only two of the signs surveyed here justify a left-right division into sign atoms to be encoded separately. All of the remainder should be encoded as sign atoms, and named (consistent with received tradition) using a sign-over-sign naming structure, not a sign-followed-by-sign structure, or in some cases with an indivisible single name SIGN. In addition, it turns out that there is only a single example in which the upper and lower parts of a SIGN-OVER-SIGN structure are distinct, so that the sign name syntax should use some word taking only a single argument, not a word like "OVER" that relates two arguments. Thus a name simply SIGN PAIRED (used in the full list of signs proposed on this site) or use of the word "PARALLEL" (as in the Ellermeier / Studt naming pattern, though they diagrammatically illustrate with two sign names) would be more appropriate.

The list below gives 27 simple sign-over-sign examples, 11 sign-crossing-sign examples, with four signs appearing in each of these two ways, possibly in contrast. There are three examples which have this kind of structure only etymologically, which can be considered fused far back in the history of Cuneiform, and of course have simple atomic names in the tradition.

Among those complex signs where we might be tempted to make a left-right division, productivity of components shows clearly that four have the structure sign-over-sign, not sign-followed-by-sign, while six are without a conclusive analysis. There are two examples of a higher complexity where we can make a first left-right division and then treat the parts as having an upper-lower division. These are (EN CROSSING EN).(IM CROSSING IM) and NINNI5.KINDA = (TIR3 OVER TIR3).((GAD.GAR) OVER (GAD.GAR)). But there are no examples of the lower degree of complexity whose major division is left-right. Since there are no such, it is probably best that the small residue of hard cases either be named as atoms or be named in the sign-over-sign structure, if parts can be identified which do function as independent signs.

Simple Examples of (SIGN OVER SIGN)

(Source?) (SU OVER SU)
Labat #6 ZU or #7 SU, the two later merged.

(source?) (BAL OVER BAL)
Labat #9 BAL, Fara #19

ZATU sign (U$ OVER U$)
ZATU sign #604 USH, Labat #50, Fara #58b

Labat #66 (ZI OVER ZI), Fara #469, ZATU sign #420
Labat #84 ZI, Fara #468, ZATU sign #617

Labat #66a (ZI OVER ZI).A
Labat #66b (ZI OVER ZI).SHE3
Labat #66c (ZI OVER ZI).LAGAB

ZATU #42 (BAD OVER BAD)
Labat #69a BAD, Fara #16, ZATU sign #41

CDLI-Civil (KAL OVER KAL)
Labat #79b KAL, Fara #709b (see Krebernik 1998 for distinction from 709a)

Labat #96a MÉNBULUG = (BULUG OVER BULUG)
Labat #96 BULUG, ZATU sign #61

Labat #99a1 (EN CROSSING EN), Fara #534
Labat #99 EN, Fara #530, ZATU sign #134

CDLI-Civil (SAG OVER SAG)
Labat #115 SAG, Fara #307, ZATU sign #437

Labat #151a (LUGAL OVER LUGAL)
Labat #151 LUGAL, Fara #165, ZATU sign #334

Labat #206a LAH = (DU OVER DU)
Labat #206 DU, Fara 484, ZATU sign #82

(Source?) (NI OVER NI)
Labat #231 NI, Fara #506, ZATU sign #393

CDLI-Civil (GA2 OVER GA2)
Labat #233 GA2, Fara #674, ZATU sign #162)

CDLI-Civil (GISH CROSSING GISH) (if at right angles, is this a name for another sign?)
Labat #296 GISH, Fara #673, ZATU sign #223

Ellermeier #330_60v (LU2 OVER LU2)
Labat #330 LU2, Fara #337, ZATU sign #332

Fara #364 (SHAR2 x DISH) OVER (SHAR2 x DISH)
Fara #373 SHAR2 x DISH

Ellermeier #375_45 NINNI5 = (TIR OVER TIR), Labat #375c, Fara #205
Labat #375b TIR, Fara #204

Labat #404x GINGIRA = (KISIM5 OVER KISIM5)
Labat #404x KISIM5

Fara #405 (KIR2 OVER KIR2)
Labat #424a KIR2, Fara #404

ZATU sign (AMAR OVER AMAR)
ZATU sign #29 AMAR, Labat #437, Fara #267

Labat #459av (DU6 OVER DU6).SHE.SAR
Labat #459a DU6, Fara 302

(source?) (GABA OVER GABA)

UET-II #174 (?MI.MI or MI OVER MI ?)

Labat #565 LUM, Fara #194, ZATU sign #335
Labat #566a (LUM OVER LUM)
Labat #566 (LUM OVER LUM).SHU2

CDLI-Civil (GUD OVER GUD)
Labat #397 GUD, Fara #123, ZATU #234

Both (SIGN OVER SIGN) and (SIGN CROSSING SIGN), in contrast?

ZATU .eps image (GI OVER GI)
Labat #67 GIL = (GI CROSSING GI) (or GI OVER GI ?), Fara #465
Labat #85 GI, Fara #464, ZATU sign #204

CDLI-Civil (GAN2 OVER GAN2)
CDLI-Civil (GAN2 CROSSING GAN2)
Labat #105a GAN2, Fara #89, ZATU sign #195

Labat #326v (GI4 OVER GI4), Fara #467, ZATU .eps image
Labat #326a (GI4 CROSSING GI4)
Labat #326 GI4, Fara #466, ZATU sign #212

Fara #234 (BU OVER BU)
Labat #371n (BU CROSSING BU).A.NA
Labat #371 BU, Fara #232

Examples of (SIGN CROSSING SIGN)

Labat #99a2 (EN CROSSING EN).(IM CROSSING IM)
Labat #99a1 (EN CROSSING EN) does exist. Although (IM CROSSING IM) may exist in Labat #399x, that sign does not have the same form as what appears in Labat #99a2. It might be instead IM OVER IM. It is conceivable that (EN.IM) exists. If it does, it might not be entered in any sign list as an "atom".

Labat #169 DAH = (MU CROSSING MU), Fara #15, ZATU sign #66
Labat #61 MU, Fara #14, ZATU sign #363

Labat #371a SIRSIR looks as if it might be a special fused form of one sign over or crossing itself.

Labat #374 RI8 = (MU$ CROSSING MU$), Fara #236
Same plus .A.NA
Labat #374 MU$, Fara #235

Labat #383a TALTAL = PI CROSSING PI
Labat #383 PI, Fara #447, ZATU sign #203a

Labat #399x (IM CROSSING IM) (because of merger, perhaps better: NI2 CROSSING NI2)
Same plus .A.NA
Labat #399 IM, Fara #377 (or better: Labat #399 NI2, Fara #376)

Labat #569 SUH3 = (GU CROSSING GU), Fara #510
Labat #559 GU, Fara #509, ZATU sign #232

Etymological examples which were no longer transparent SIGN OVER SIGN
Many other examples listed here were also surely no longer transparent)

NIR = *(NUN OVER NUN)
NAB = *(AN OVER AN) (and MUL = cluster of three stars, not NAB.AN)
SUM = *(SHE OVER SHE)

Examples of (SIGN OVER SIGN) where an attempt to divide the total horizontally as (Component.Component) is not supported by listed signs.
In these signs, a first-level vertical division into (Component OVER Component) does work, and the sign can be named in that way using the names of existing signs as the components.

Labat #441a (UL OVER UL)
Labat #441 UL, Fara #300 (& 299 ?)
Although UL appears like U x GUD, a sign (U OVER U) is not listed as an independent sign, wheras UL does exist.

(source?) (KAD5 OVER KAD5)
Labat #206 KAD5, Fara #173
Although KAD5 appears like SHU.KAD3, neither (SHU OVER SHU) nor (KAD3 OVER KAD3) are listed as independent signs.

Ellermeier #63_27 (TUR3 OVER TUR3)
Labat #87a TUR3
Although TUR3 looks like NUN.LAGAR, neither (NUN OVER NUN) nor (LAGAR OVER LAGAR) are listed as independent signs. A sign NIR does exist which can be traced etymologically to NUN OVER NUN. It is Labat #325, Fara #122, ZATU sign #414. But this form is not the same as what appears in (NUN.LAGAR) over (NUN.LAGAR).

Labat #87c ($ILAM OVER $ILAM)
Labat #87b $ILAM = TUR3 x SAL
Again, $ILAM appears like NUN.(LAGAR x SAL), but neither (NUN OVER NUN) nor ((LAGAR x SAL) OVER (LAGAR x SAL)) are listed as independent signs. This example is also like the some of the signs for insects, in that the structural division between container sign component and infixed sign component reveal the container to be visually rather long left-to-right. While it would be tempting to divide the container sign, such a division yields non-productive components, not supported elsewhere.

Examples of (SIGN OVER SIGN) where neither way of making a first division is supported by listed signs.

Labat #567a appears like (SIG4 OVER SIG4).SHU2.
But (SIG4 OVER SIG4) is not listed as an independent sign.

Ellermeier #449_46 DIMSAR ?? = (IGI.SHIR.gunu.PUR2) OVER (IGI.SHIR.gunu.PUR2). All of the three elements exist independently, but none of them doubled as SIGN OVER SIGN is listed as an independent sign.

Labat #369 (SHE.GAD.GAR) OVER (SHE.GAD.GAR).
The combination ((GAD.GAR) OVER (GAD.GAR)) does recur as KINDA Labat #91, and LAK indicates it derives from an earlier
(U.SUR). While (SHE OVER SHE) is not listed as an independent sign, what was etymologically (SHE OVER SHE) can be found in a very different form in Labat #64 SUM, Fara 200, ZATU sign #515. See the etymological examples at the end. Pursuing (U.SUR) does not get us farther towards productivity.

Labat #370 (SHE.TAB.GAR) OVER (SHE.TAB.GAR)
TAB.GAR is not listed as an independent sign, nor is its doubled stacked version. Nor is SHE.TAB.GAR. See the preceding item.

Labat #566b (LUM.GAR) OVER (LUM.GAR)
or (LUM OVER LUM).(GAR OVER GAR)
Neither analysis for this item finds support, since neither (LUM.GAR) nor (GAR OVER GAR) are listed as independent signs, even though (LUM OVER LUM) does exist.

Labat #144f ZIZNA = (TUR.ZA OVER TUR.ZA)
None of these three are listed as independent signs: not TUR.ZA, nor (TUR OVER TUR), nor (ZA OVER ZA). So there is no productivity basis for choosing how to divide the four-part sign.

Problem examples (suggest simply coding and naming as single signs, at least until a better analysis is pointed to?)

Ellermeier #375_46 NINNI5.KINDA = (TIR OVER TIR).KINDA.
Because we have the single name KINDA, we can name this one via its two major parts, the first of them sign-over-sign. For KINDA, any attempt to analyze further faces a lack of functional parts, so we need four sign names and nested parenthses: ((GAD.GAR) OVER (GAD.GAR)). It is one of only two which have a structurally supported primary division left-right instead of above-below.

Labat #125a UBISHAGA (is this a deformation of an unrecognized sign when stacked one above another?)

Labat #99b2 MERMER5 (unknown parts, and a completely unique arrangement ((X.X) over (Y.Y))

Labat #99b3 GASHMU looks like (ASH.TUG2.TUG2.ASH) CROSSING (ASH.TUG2.TUG2.ASH)

HILIBU (? source for listing ?) appears to be (SHU.SHU.tenu) over (SHU.SHU.tenu),
but neither (SHU OVER SHU) nor (SHU.tenu OVER SHU.tenu) exist independently?
Pehaps (SHU OVER SHU.inverted) does exist ? (? source for listing ?)