  Let $G$ (resp. $H$) be the group of orientation preserving
self-homeomorphisms of the unit circle (resp. real line). In previous work, the
first two authors constructed pre-Tannakian categories
$\underline{\mathrm{Rep}}(G)$ and $\underline{\mathrm{Rep}}(H)$ associated to
these groups. In the predecessor to this paper, we analyzed the category
$\underline{\mathrm{Rep}}(H)$ (which we named the ``Delannoy category'') in
great detail, and found it to have many special properties. In this paper, we
study $\underline{\mathrm{Rep}}(G)$. The primary difference between these two
categories is that $\underline{\mathrm{Rep}}(H)$ is semi-simple, while
$\underline{\mathrm{Rep}}(G)$ is not; this introduces new complications in the
present case. We find that $\underline{\mathrm{Rep}}(G)$ is closely related to
the combinatorics of objects we call Delannoy loops, which seem to have not
previously been studied.
