  This paper contains the classification of the orbits of elements of the
tensor product spaces ${\mathbb{F}}^2\otimes {\mathbb{F}}^3
\otimes{\mathbb{F}}^r$, $r\geq 1$, under the action of two natural groups, for
all finite; real; and algebraically closed fields. For each of the orbits we
determine: a canonical form; the tensor rank; the rank distribution of the
contraction spaces; and a geometric description. The proof is based on the
study of the contraction spaces in
${\mathrm{PG}}({\mathbb{F}}^2\otimes{\mathbb{F}}^3)$ and is geometric in
nature. Although the main focus is on finite fields, the techniques are mostly
field independent.
