Semideviation in Sharpe or Beta?

Why can’t you use semideviation in sharpe or beta calculations?

The Sharpe ratio is just defined that way. The Sortino ratio uses downside deviation if you want to use that.

Beta is a covaraiance measure. There are upside and downside betas possible if you want to do the regressions to get them. Usually, they would involve an interacting dummy variable for market direction.