  We present a class of inflationary potentials which are invariant under a
special symmetry, which depends on the parameters of the models. As we show, in
certain limiting cases, the inverse symmetric potentials are qualitatively
similar to the $\alpha$-attractors models, since the resulting observational
indices are identical. However, there are some quantitative differences which
we discuss in some detail. As we show, some inverse symmetric models always
yield results compatible with observations, but this strongly depends on the
asymptotic form of the potential at large $e$-folding numbers. In fact when the
limiting functional form is identical to the one corresponding to the
$\alpha$-attractors models, the compatibility with the observations is
guaranteed. Also we find the relation of the inverse symmetric models with the
Starobinsky model and we highlight the differences. In addition, an alternative
inverse symmetric model is studied and as we show, not all the inverse
symmetric models are viable. Moreover, we study the corresponding $F(R)$
gravity theory and we show that the Jordan frame theory belongs to the $R^2$
attractor class of models. Finally we discuss a non-minimally coupled theory
and we show that the attractor behavior occurs in this case too.
