  The nonorientable four-ball genus of a knot $K$ in $S^3$ is the minimal first
Betti number of nonorientable surfaces in $B^4$ bounded by $K$. By amalgamating
ideas from involutive knot Floer homology and unoriented knot Floer homology,
we give a new lower bound on the smooth nonorientable four-ball genus
$\gamma_4$ of any knot. This bound is sharp for several families of torus
knots, including $T_{4n,(2n\pm 1)^2}$ for even $n\ge 2$, a family Longo showed
were counterexamples to Batson's conjecture. We also prove that, whenever $p$
is an even positive integer and $\frac{p}{2}$ is not a perfect square, the
torus knot $T_{p,q}$ does not bound a locally flat M\"obius band for almost all
integers $q$ relatively prime to $p$.
