A bit like decathlon/heptathlon.
If there is an interest, they could split the two "gymnastics", give them different names, and have both genders compete in both disciplines. In the same way that there is a women's decathlon (but not in the Olympics).
Mental games are wide open, but there's a reason physical games are segregated. Evolution built us to be good at different things.
You could use gendered world records and averages to weight individual scores in competitions, allowing desegregated competition, essentially giving women a handicap. There's a chance that the psychology of competition could elicit better performance and even things out, or that competing with men directly could negatively impact performance.
Politics du'jour isn't very compatible with reality in some areas. Historically speaking, the strongest and fastest or most endurant person on the planet is going to be male. I don't think it's fair to women to have to compete against the realities of biology without fundamentally changing the way things are being measured. And it's not fair to men to limit the measurement of performance on a direct and literal scale.