  We present models where $\gamma_+$ and $\gamma_-$, the exponents of the
susceptibility in the high and low temperature phases, are generically
different. In these models, continuous symmetries are explicitly broken down by
discrete anisotropies that are irrelevant in the renormalization-group sense.
The $\mathbb{Z}_q$-invariant models are the simplest examples for two-component
order parameters ($N=2$) and the icosahedral symmetry for $N=3$. We compute
accurately $\gamma_+ -\gamma_-$ as well as the ratio $\nu/\nu'$ of the
exponents of the two correlation lengths present for $T<T_c$.
