We study a two-parameter family of planar difference equations obtained from the classical Hénon map by letting the delay of the quadratic term depend on the sign of the current state: the term $x_n^2$ is used when $x_n \ge 0$ and $x_{n-1}^2$ when $x_n < 0$. The resulting map is discontinuous on the switching line $x_n = 0$ and non-invertible. We prove that the two Hénon fixed points persist but that the saddle one becomes a repeller, that the stable 2-cycle of the Hénon map persists exactly up to $a^* = (1-b)^2$, and that no 2-cycle exists beyond this value. Numerical experiments for $b=0.3$ show that, beyond $a^*$, the attractor is a sequence of stable odd-period cycles whose period decreases as $a$ grows, followed by chaotic regimes, and that several attractors (for instance a 5-cycle and a 10-cycle at $a=0.9$) coexist with finely interwoven basins of attraction. For $b=0.3$ we prove, with explicit tail estimates, that stable cycles of every odd period $N \ge 5$ exist on intervals of $a$ that accumulate at $a^*$ geometrically.