  Time reversal symmetric triangular maps of the unit square are introduced
with the property that the time evolution of one of their two variables is
determined by a piecewise expanding map of the unit interval. We study their
statistical properties and establish the conditions under which their
equilibrium measures have a product structure, i.e. factorises in a symmetric
form. When these conditions are not verified, the equilibrium measure does not
have a product form and therefore provides additional information on the
statistical properties of theses maps. This is the case of anti-symmetric cusp
maps, which have an intermittent fixed point and yet have uniform invariant
measures on the unit interval. We construct the invariant density of the
corresponding two-dimensional triangular map and prove that it exhibits a
singularity at the intermittent fixed point.
