Summary: An orientably-regular map is a 2-cell embedding of a connected graph or multigraph into an orientable surface, such that the group of all orientation-preserving automorphisms of the embedding has a single orbit on the set of all arcs (incident vertex-edge pairs). Such embeddings of the \(n\)-dimensional cubes \(Q_{n}\) were classified for all odd \(n\) by S. F. Du, J. Ho and R. Nedela [Eur. J. Comb. 26, No. 3–4, 505–519 (2005; Zbl 1057.05026)], and in 2007 by J. Xu [Sci. China, Ser. A 50, No. 12, 1673–1679 (2007; Zbl 1133.05023)] proved that for \(n=2m\) where \(m\) is odd, they are precisely the embeddings constructed by Y. S. Kwon [J. Graph Theory 46, No. 4, 297–312 (2004; Zbl 1046.05022)]. Here, we give a classification of orientably-regular embeddings of \(Q_{n}\) for all \(n\). In particular, we show that for all even \(n (=2m)\), these embeddings are in one-to-one correspondence with elements \(\sigma \) of order 1 or 2 in the symmetric group \(S_{n}\) such that \(\sigma \) fixes \(n\), preserves the set of all pairs \(B_{i}=\{i,i+m\}\) for \(1\leq i\leq m\), and induces the same permutation on this set as the permutation \(B_{i} \rightarrowtail B_{f(i)}\) for some additive bijection \(f:\mathbb Z_{m}\rightarrow \mathbb Z_{m}\). We also give formulae for the numbers of embeddings that are reflexible and chiral, respectively, showing that the ratio of reflexible to chiral embeddings tends to zero for large even \(n\).