How about players getting a game each as white and black?
Proposal to equalize black...

How about players getting a game each as white and black?
Simultaneous games against a given opponent...one as white...one as black? Interesting. I've only had one cup of coffee so far this morning but it would seem to be a logical evolution.

Sometimes having an opening reversed/with an extra tempo turns out to be disadvantageous -- and sometimes only advantageous only if you're very careful with the move order / how you spend the extra tempo.
I would hardly call 1.c3 a reversed caro or 1.e3 a reversed pirc/modern. After this one ply the opening could transpose into many things, you'd have to go deeper into each game to classify it's opening.
The results from games with e.g. 1.c3 should be taken with a grain of salt for their # and the strength of the players who opened this way.

As an example, let's look at 1. c3 and try to understand the statistics.
If black plays 1. ... e5 white scores 42% - 27% - 31% or 56%/44%
The Caro-Kann doesn't score nearly this well for black. So the Caro with an extra tempo is good.
The problem is that black doesn't have to play 1. ... e5. White does worse against other black moves. So we can't really consisder 1. ... c3 as a tempo up Caro-Kann unless black cooperates.
Also, as you point out, the samples are pretty small in the games explorer. 26 games in the 1. ... e5 line. No more than 58 games in any other line.
Another option for evening out the playing field is having one player make a move for white and then having the other player decide which side they want to play.

Everybody knows that a game between equally highly skilled players is biased towards white because it has a free tempo.
One way to counter this could be to say "Ok , you white can move first, but you cannot claim the center right away"
...
I think that the success of hypermodern openings, even when facing classical opposition that does claim the centre, is an indication that choosing not to lay a direct claim in the centre on the first move isn't really a factor that would equalize any advantage conferred as a result of being able to move first.
As suggested above, having each side play one game as Black and another as White is a far better way to ensure equality.

I've never heard of this, I like that idea.

Another option for evening out the playing field is having one player make a move for white and then having the other player decide which side they want to play.
This is a really good one ... Which moves would you choose yourself?

I've never heard of this, I like that idea.
It is interesting. It would introduce a new dimension to strategy (mind games).

Interesting -- "I cut, you choose."
Mightn't it be better yet to have the first player make a move for white, and then have the second player decide whether he wants to choose a side, or play a response for black and throw the choice back to the first player?

Something that would allow an undefined number of ply before someone chooses would be interesting. But...I got confused even thinking about it.

Suppose white makes the 1st & 4th moves (and even-numbered moves thereafter), and black makes the 2nd & 3rd moves (and odd-numbered moves thereafter)?

How about players getting a game each as white and black?
Simultaneous games against a given opponent...one as white...one as black? Interesting. I've only had one cup of coffee so far this morning but it would seem to be a logical evolution.
Didn't quite say it was simultaneous. Just that after A plays B, they play another with switched colors. It's a crazy idea, but it just might work :)

Looking at the game explorer the more balanced openings are b3 and the orangutang b4, but opening knowledge has to be taken in account I suppose

I'm thinking of the notion that best play ends in a draw. Something where less than best play might be a strategy.
Nevermind. I can just see games where each player makes the worse move they can think of. Then again, I might excel.
Everybody knows that a game between equally highly skilled players is biased towards white because it has a free tempo.
One way to counter this could be to say "Ok , you white can move first, but you cannot claim the center right away"
A way to achieve this could be to allow on the first white move only 1 step pawn advances (and no piece move). In this way theoretically black should become white and white should become black with an extra tempo which should equalize the game sufficiently.
I tried to analyize the openings outcomes in the game explorer and the results are disconcerting:
g3 = modern with extra tempo : favors white (+12% win)
e3 = french with extra tempo
d3 = pirc with extra tempo
and
c3 = caro kann with extra tempo : all favor black (+12%,+4%,+6%)
So I'd like to ask to the fellow forumnites if they can enlighten me: am I missing something or are the samples relative to the uncommon openings in the game explorer unreliable ?
Maybe some black defences with extra tempo are simply bad ...
Regards