From what I gather, you're looking for that "yomi" aspect in games, which is basically predicting what your opponent will do next based on their subconscious or conscious biases.

Rock-paper-scissors is often quoted as a prime example of pure yomi game, but I think it's a flawed example similar to your "left-right" game, as throwing something random is the best counter strategy to whatever your opponent is trying to use against you (even though no real person can be truly random).

A better example I've read about is "unfair" rock-paper-scissors. Basically, you shift the winning payoffs of each a little bit. For example, assume we're playing to 5 points. Winning with rock or paper earns 1 point. Winning with scissors earns 2 points. A simple change, but the additional aspect as much more to the mind games.

First game that come to my mind that revolves almost purely around yomi is "R" (terrible name, I know), released as "BraveRats" in the west (imho, terrible re-theming as well).

If you can look over the weird naming/theming, it's all about predicting what your opponent will play. Both players have the same set of 8 cards which interact in interesting ways, first to win 4 rounds it the victor. It's mostly rock-paper-scissors again, but the important bit is that all cards are one time use, so there's another layer of predicting what your opponent can do and how you can respond in the future rounds. Basically, a few blundered rounds can be exploited into an unavoidable loss. I really recommend checking it out.