cool idea with the “kissing sides”!
Thanks for sharing! It’s great how this enhances the tactile part of rolling dice, instead of just reducing them to little random number generators.
You mention the (lack of) independence from the main result, but it’s a little more interesting than that, so I’ll describe it here in case it’s useful for you. The post says:
These ‘side doubles’ are more or less independent of the main result on the top faces, i.e. about 1 in 6* (although if the top faces are a double, then the probability rises (I believe) to 1 in 4, which seems app to my mind).
But actually (TL;DR):
On most d6s, if the main result shows doubles or sums to 7, there’s a 1 in 4 chance of getting side doubles. Otherwise, there’s a 1 in 8 chance.
In more detail:
Many (most?) d6s have paired sides: 1 and 6 are on opposite faces, as are 2/5 and 3/4. This means that rolling 1 and 6 reveals the same four sides that might get paired into side doubles: 2, 3, 4, 5. There are 16 ways to click the dice together and with those sides available, four of them are doubles, so that’s 1 in 4 or 25% of the time if the main result is doubles or opposite faces (summing to 7).
With any other main results, the four sides that might get paired into side doubles will be different between the two dice. On a roll of 1 and 5 for instance, you’ve got {2, 3, 4, 5} x {1, 3, 4, 6}. Out of the 16 combinations, only two are side doubles (33 and 44), so that’s a 1 in 8 chance or 12.5%.
Two nice results of this:

If you’re playing with someone who’s using digital dice, they can recreate side doubles with one extra roll:
If the main result is doubles or sums to 7, roll a d4. Otherwise, roll a d8. Either way, if the result is 1, you scored a “side double.”

I think your rule is more interesting, but if you want truly independent doubles, it’s still the case that the overall odds of doubles is 1 in 6, so you can do something like:
Alongside your main roll, roll a third d6 of a different color. If the result is 1, you scored a “side double.”
Depending on what getting a double means in the context of your game, this could be tactile in its own way—that third die could represent “luck” or “chaos.”
Nice  will give this a think over!
:O)
I missed the idea that the ‘inverse’ of a double (AKA top faces ‘sum to 7’) also has a 1 in 4 of giving a ‘side double’! Or rather, I failed to follow this insight through to the conclusion!
:O)
Since you are good at this, am I correct is saying:
1:: any ‘top double’ is 1 in 6
2:: any ‘side double’ is 1 in 6
3:: a ‘side double’ without a top double is … (hrmmm)
4:: a ‘top double’ without a side double is … 1/6 * 3/4 = 1/8
5:: a ‘top double’ + ‘side double’ = 1/6 * 1/4 = 1 in 24,
:O/
Your conclusions (1, 2, 4, 5) look right.
3:: a ‘side double’ without a top double is … (hrmmm)
This is equal to [the likelihood of a top inverse with a side double] plus [the likelihood of a top “other” (not double or inverse) with a side double]. That’s:
TiSd = p(top_inverse) * p(side_double  top_inverse) = 1/6 * 1/4 = 1/24
ToSd = p(top_other) * p(side_double  top_other) = 2/3 * 1/8 = 1/12
TiSd + ToSd = 1/24 + 1/12 = 1/8
Here’s a summary of the possible combinations:
Top Double  Top Inverse  Top Other  Total  

Side Double  1/24  1/24  1/12  1/6 
Side Other  1/8  1/8  7/12  5/6 
Total  1/6  1/6  2/3  1 
Maybe most readable with a common denominator:
Top Double  Top Inverse  Top Other  Total  

Side Double  1/24  1/24  2/24  4/24 
Side Other  3/24  3/24  14/24  20/24 
Total  4/24  4/24  16/24  24/24 
This is pretty interesting and something I’ll have to look into more intently for my home games and the like.
Reading @gravenutterance’s replies, I knew there was a bell curve involved with the 2d6, but never really thought about the extra layer of possible events that would occur with doubles and the like. The showcased math of their work warrants special events to occur.