SCOTUS reaffirmed the State's right to present rebuttal expert testimony based on an interview of the defendant if the defendant initiates such evidence.

Details.

This has been the law in Texas under Lagrone v. State, 942 SW2d 602, since 1997. Interestingly, though, SCOTUS did not talk about whether the justification arose from a concept of waiver of the 5th Amendment. Instead, the Court focused on the fundamental unfairness of allowing the Defendant to present only one side to a mental health issue.

Does that mean the State has some fundamental right to a fair trial?

Seems to at least mean the defendant does not have a right to an unfair trial on an issue on which he bears the burden of proof. The ruling by the Kansas Supreme Court was a bit frightening: take enough meth and you can do anything you want without criminal sanctions.