  A non-zero element of the Lie algebra $\mathfrak{se}(3)$ of the special
Euclidean spatial isometry group $SE(3)$ is known as a {\em twist} and the
corresponding element of the projective Lie algebra is termed a {\em screw}.
Either can be used to describe a one-degree-of-freedom joint between rigid
components in a mechanical device or robot manipulator. This leads to a
practical interest in multiple twists or screws, describing the overall
instantaneous motion of such a device. In this paper, invariants of multiple
twists under the action induced by the adjoint action of the group are
determined. The ring of the polynomial invariants for the adjoint action of
$SE(3)$ acting on a single twist is well known to be finitely generated by the
Klein and Killing forms, while a theorem of Panyushev gives finite generation
for the real invariants of the induced action on two twists. However we are not
aware of a corresponding theorem for $k$~twists, where $k\geq3$. Following
Study, we use the principle of transference to determine fundamental algebraic
invariants and their syzygies. We prove that the ring of invariants for triple
twists is rationally finitely generated by $13$ of these invariants.
