Hyperbolic X-piece ($X({\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4;s,\tau )$)

An X-piece is a curvature-$-1$ four-holed sphere with geodesic boundary, formed by gluing two Y-pieces along equal-length cuffs. For a specified separating cuff and labelled outer boundaries, four outer lengths, the internal cuff length $s>0$, and a twist $\tau$ determine the metric. The gluing is orientation-reversing on boundary circles, with relative arclength displacement $\tau$ measured from chosen seam endpoints. Without markings, the gluing displacement is taken modulo $s$; with a marking, $\tau \in \mathbb{R}$ records Dehn twists as well. Changing the chosen separating curve can give different parameter descriptions of the same unmarked surface.