  We investigate symmetry properties of peculiar modules, a Heegaard Floer
invariant of 4-ended tangles which the author introduced in [arXiv:1712.05050].
In particular, we give an almost complete answer to the geography problem for
components of peculiar modules of tangles. As a main application, we show that
Conway mutation preserves the hat flavour of the relatively $\delta$-graded
Heegaard Floer theory of links.
