  A QRT map is the composition of two involutions on a biquadratic curve: one
switching the $x$-coordinates of two intersection points with a given
horizontal line, and the other switching the $y$-coordinates of two
intersections with a vertical line. Given a QRT map, a natural question is to
ask whether it allows a decomposition into further involutions. Here we provide
new answers to this question and show how they lead to a new class of maps, as
well as known HKY maps and quadrirational Yang-Baxter maps.
