On symmetries of peculiar modules, or $\delta$-graded link Floer homology is mutation invariant

We investigate symmetry properties of peculiar modules, a Heegaard Floer invariant 4-ended tangles which the author introduced in \[J. Topol. 13 (2020)]. In particular, we give an almost complete answer to geography problem for components modules tangles. As main application, show that Conway mutation preserves hat flavour relatively $\delta$-graded theory links.