Fact. Given $ABC$, a circumhyperbola $\ell^*$ is rectangular (orthogonal asymptotes) iff it passes through $H$.

Proof. The isogonal conjugate $\ell$ is a line, and the intersections $\ell \cap (ABC)$ are the isogonal conjugates of the infinity points along the asymptotes. So the asymptotes were orthogonal iff the intersections are diametrically opposite, or that $\ell$ passes through $O$. Flipping this we get $\ell^*$ passes through $H$.

Fact. Chord $AH$ of a rectangular hyperbola $\mathcal H$ passes through its center. Then on $\mathcal H$, the map $B\mapsto C$ where $C$ is the orthocenter of $AHB$ is an involution.

Proof. Clearly, $B\mapsto C \mapsto B$. Projectivity is immediate since $AB\perp HC$.

Fact. (cotinued from previous) If $\{X,Y\} = (AH) \cap \mathcal H$, then for any $P\in \mathcal H$, $PX$ and $PY$ bisect $\angle BPC$.

Proof. $A',H'$ are the fixed point of the involution, since $\angle AA'H=\angle AH'H = 90^\circ$ and the orthocenter of a right-angled triangle is just the vertex with the right angle.