  We define an action of the operad of projective spineless cacti on each stage
of the Taylor tower for the space of framed 1-dimensional long knots in any
Euclidean space. By mapping a subspace of the overlapping intervals operad to
the subspace of normalized cacti, we prove a space-level compatibility of our
action with Budney's little 2-cubes action on the space of framed long knots
itself. Our result improves upon previous joint work of the first author
related to the conjecture that the Taylor tower for classical long knots is a
universal Vassiliev invariant over the integers. As a corollary, we reprove the
nontriviality of a certain Browder bracket class first detected by Sakai.
