  Let $G$ be the group of all order-preserving self-maps of the real line. In
previous work, the first two authors constructed a pre-Tannakian category
$\underline{\mathrm{Rep}}(G)$ associated to $G$. The present paper is a
detailed study of this category, which we name the Delannoy category. We
classify the simple objects, determine branching rules to open subgroups, and
give a combinatorial rule for tensor products. The Delannoy category has some
remarkable features: it is semi-simple in all characteristics; all simples have
categorical dimension $\pm 1$; and the Adams operations on its Grothendieck
group are trivial. We also give a combinatorial model for
$\underline{\mathrm{Rep}}(G)$ based on Delannoy paths.
