  We prove new Fourier restriction estimates to the unit sphere $S^{d-1}$ on
the class of $O(d-k)\times O(k)$-symmetric functions, for every $d\geq 4$ and
$2\leq k\leq d-2$. As an application, we establish the existence of maximizers
for the endpoint Stein--Tomas inequality within that class. Moreover, we
construct examples showing that the range of Lebesgue exponents in our
estimates is sharp.
