We fully describe the horofunction boundary δhL2 with the word metric associated with the generating set {t, at} (i.e. the metric arising in the Diestel–Leader graph DL(2, 2)). The visual boundary δ∞L2 with this metric is a subset of δhL2. Although δ∞L2 does not embed continuously in δhL2, it naturally splits into two subspaces, each of which is a punctured Cantor set and does embed continuously. The height function on DL(2, 2) provides a natural stratification of δhL2, in which countably many non-Busemann points interpolate between the two halves of δ∞L2. Furthermore, the height function and its negation are themselves non-Busemann horofunctions in δhL2 and are global fixed points of the action of L2.