The phenomenon of fractional self-imaging can be used to implement atom-optical components in the interior of a rectangular quantum billiard. Their temporal concatenation may lead to complex "atom chips" operating in the time-domain. As an elementary example, we discuss a billiard internal Mach-Zehnder interferometer that can be transformed into an all-optical switch or controllable beam splitter for atoms. The inherent coherence and reversibility of its dynamics also provides a simple means to realize a Welcher Weg/Quantum erasure experiment. The whole concept is shown to be robust with respect to realistic wall-softness effects and feasible within the present state of the art.