  The recent observation of dipole-allowed $P$-excitons up to principal quantum
numbers of $n=25$ in cuprous oxide has given insight into exciton states with
unprecedented spectral resolution. While so far the exciton description as a
hydrogen-like complex has been sufficient for cubic crystals, we demonstrate
here distinct deviations: The breaking of rotational symmetry leads to mixing
of high angular momentum $F$- and $H$-excitons with the $P$-excitons so that
they can be observed in absorption. The $F$-excitons show a three-fold
splitting that depends systematically on $n$, in agreement with theoretical
considerations. From detailed comparison of experiment and theory we determine
the cubic anisotropy parameter of the Cu$_2$O valence band.
