  We study the critical behavior of a general class of cubic-symmetric spin
systems in which disorder preserves the reflection symmetry $s_a\to -s_a$,
$s_b\to s_b$ for $b\not= a$. This includes spin models in the presence of
random cubic-symmetric anisotropy with probability distribution vanishing
outside the lattice axes. Using nonperturbative arguments we show the existence
of a stable fixed point corresponding to the random-exchange Ising universality
class. The field-theoretical renormalization-group flow is investigated in the
framework of a fixed-dimension expansion in powers of appropriate quartic
couplings, computing the corresponding $\beta$-functions to five loops. This
analysis shows that the random Ising fixed point is the only stable fixed point
that is accessible from the relevant parameter region. Therefore, if the system
undergoes a continuous transition, it belongs to the random-exchange Ising
universality class. The approach to the asymptotic critical behavior is
controlled by scaling corrections with exponent $\Delta = - \alpha_r$, where
$\alpha_r\simeq -0.05$ is the specific-heat exponent of the random-exchange
Ising model.
