Canonical orientations in Heegaard Floer theory

We set up Heegaard Floer theory over the integers, using canonical orientations coming from coupled Spin structures on the Lagrangian tori. We prove naturality of Heegaard Floer homology, sutured Floer homology, and link Floer homology over $\mathbb{Z}$. We give a new proof of the surgery exact triangle in this context, as well as a definition of involutive Heegaard Floer homology over $\mathbb{Z}$.