  We explore the effect of valley-contrasting gaps in the optical response of
two-dimensional anisotropic tilted Dirac systems. We study the spectrum of
intraband and interband transitions through the joint density of states (JDOS),
the optical conductivity tensor, and the Drude spectral weight. The energy
bands present an indirect gap in each valley. Thus a new possibility opens for
the position of the Fermi level (an ``indirect zone''), and for the momentum
space for allowed transitions. The JDOS near each gap displays a set of three
van Hove singularities which are in contrast to the case of gapped graphene (an
absorption edge only) or 8-$Pmmn$ borophene (two interband critical points due
to the tilt). For the Fermi level lying in the gap, the JDOS shows the usual
linear dependence on frequency, while when above an indirect zone it looks
similar to the borophene case. These spectral characteristics determine the
prominent structure of the optical conductivity. The longitudinal conductivity
illustrates the strong anisotropy of the optical response. Similarly, the Drude
weight is anisotropic and shows regions of nonlinear dependence on the Fermi
level. The breaking of valley symmetry leads to a finite Hall response and
associated optical properties. The anomalous and valley Hall conductivities
present graphene-like behavior with characteristic modifications due to the
indirect zones. Almost perfect circular dichroism can be achieved by tuning the
exciting frequency with an appropriate Fermi level position. We also calculate
the spectra of optical opacity and polarization rotation, which can reach
magnitudes of tenths of radians in some cases. The spectral features of the
calculated response properties are signatures of the simultaneous presence of
tilt and mass and suggest optical ways to determine the formation of different
gaps in such class of Dirac systems.
