  We study an assembly-type bias parametrized by the dimensionless spin
parameter that affects massive structures. In numerical simulations higher spin
haloes are more strongly clustered than lower spin haloes of equal mass. We
detect a difference of over a 30 per cent in the clustering strength for dark
matter haloes of 10^13-10^14 Msun, which is similar to the result of Bett et
al. We explore whether the dependence of clustering strength on halo spin is
removed if we apply the redefinition of overdensity peak height proposed by
Lacerna & Padilla (Paper I) obtained using assembly ages. We find that this is
not the case due to two reasons. Firstly, only a few objects of low-virial mass
are moved into the mass range where the spin introduces an assembly bias after
using this redefinition. Secondly, this formalism does not alter the mass of
massive objects. We then repeat the process of finding the redefined peak
height of Paper I but using the spin. In this case, the new masses show no
spin-related assembly bias but they introduce a previously absent assembly bias
with respect to relative age. From this result, we conclude that the
assembly-type bias with respect to the halo spin has a different origin than
with respect to assembly age. The former may be due to the material from
filaments, which is accreted by massive haloes, that is enhanced in
high-density environments, thus causing more extreme spin values without
significantly changing the formation age of the halo. In addition, high-mass
objects may correspond, in some cases, to a different peak height than that
suggested by their mass in numerical simulations, providing a possible
explanation for the assembly bias with respect to spin. (abridged)
