  We lay the foundation for a version of $r$-spin theory in genus zero for
Riemann surfaces with boundary. In particular, we define the notion of $r$-spin
disks, their moduli space, and the Witten bundle, we show that the moduli space
is a compact smooth orientable orbifold with corners, and we prove that the
Witten bundle is canonically relatively oriented relative to the moduli space.
In the sequel to this paper, we use these constructions to define open $r$-spin
intersection theory and relate it to the Gelfand-Dickey hierarchy, thus
providing an analogue of Witten's $r$-spin conjecture in the open setting.
