  We consider two coupled quantum tops with angular momentum vectors
$\mathbf{L}$ and $\mathbf{M}$. The coupling Hamiltonian defines the
Feinberg-Peres model which is a known paradigm of quantum chaos. We show that
this model has a nonstandard symmetry with respect to the Altland-Zirnbauer
tenfold symmetry classification of quantum systems which extends the well-known
threefold way of Wigner and Dyson (referred to as `standard' symmetry classes
here). We identify that the nonstandard symmetry classes BD$I_0$ (chiral
orthogonal class with no zero modes), BD$I_1$ (chiral orthogonal class with one
zero mode) and C$I$ (antichiral orthogonal class) as well as the standard
symmetry class A$I$ (orthogonal class). We numerically analyze the specific
spectral quantum signatures of chaos related to the nonstandard symmetries. In
the microscopic density of states and in the distribution of the lowest
positive energy eigenvalue we show that the Feinberg-Peres model follows the
predictions of the Gaussian ensembles of random-matrix theory in the
appropriate symmetry class if the corresponding classical dynamics is chaotic.
In a crossover to mixed and near-integrable classical dynamics we show that
these signatures disappear or strongly change.
