  We study sets of mutually orthogonal Latin rectangles (MOLR), and a natural
variation of the concept of self-orthogonal Latin squares which is applicable
on larger sets of mutually orthogonal Latin squares and MOLR, namely that each
Latin rectangle in a set of MOLR is isotopic to each other rectangle in the
set. We call such a set of MOLR \emph{homogeneous}. In the course of doing
this, we perform a complete enumeration of non-isotopic sets of $t$ mutually
orthogonal $k\times n$ Latin rectangles for $k\leq n \leq 7$, for all $t < n$.
Specifically, we keep track of homogeneous sets of MOLR, as well as sets of
MOLR where the autotopism group acts transitively on the rectangles, and we
call such sets of MOLR \emph{transitive}. We build the sets of MOLR row by row,
and in this process we also keep track of which of the MOLR are homogeneous
and/or transitive in each step of the construction process. We use the prefix
\emph{stepwise} to refer to sets of MOLR with this property. Sets of MOLR are
connected to other discrete objects, notably finite geometries and certain
regular graphs. Here we observe that all projective planes of order at most 9
except the Hughes plane can be constructed from a stepwise transitive MOLR.
