Correction terms of double branched covers and symmetries of immersed curves

We use the immersed curves description of bordered Floer homology to study $d$-invariants of double branched covers ${\mathrm{\Sigma }}_2(L)$ of arborescent links $L\subset S^3$. We define a new invariant ${\mathrm{\Delta }}_{sym}$ of bordered ${\mathbb{Z}}_2$-homology solid tori from an involution of the associated immersed curves and relate it to both the $d$-invariants and the Neumann-Siebenmann $\bar{\mu }$-invariants of certain fillings. We deduce that if $L$ is a 2-component arborescent link and ${\mathrm{\Sigma }}_2(L)$ is an L-space, then the spin $d$-invariants of ${\mathrm{\Sigma }}_2(L)$ are determined by the signatures of $L$. By a separate argument, we show that the same relationship holds when $L$ is a 2-component link that admits a certain symmetry.