This paper presents a framework for learning vision-based robotic policies for contact-rich manipulation tasks, which generalize spatially across task configurations. We propose EquiContact, a novel hierarchical framework composed of a high-level vision planner, such as a Diffusion Equivariant Descriptor Field (Diff-EDF) and a low-level compliant visuomotor policy, such as a Geometric Compliant Action Chunking Transformer (G-CompACT). G-CompACT, operating using localized observations from wrist-mounted RGB images, end-effector pose feedback in the form of a geometrically consistent error vector (GCEV) and force-torque readings, produces actions defined on the end-effector frame. We show that the entire EquiContact pipeline is SE-equivariant under equivariant reference-frame estimation and left-invariant visual features, from perception to force control. Once the robot learns to execute a contact-rich task, such as inserting a peg-in-hole (PiH), it can execute similar tasks at different locations and orientations without retraining of the low-level policy. We also describe three key components for attaining spatially generalizable contact-rich policies: geometrically consistent compliance, the use of localized policies, and induced equivariance. Real-world peg-in-hole (PiH) experiments demonstrate near-perfect success rates under spatially out-of-distribution configurations, validating the proposed framework and principles, while additional screwing and surface-wiping experiments evaluate the transfer of the localized policy design beyond insertion. The experimental videos and more details can be found on the project website: https://equicontact.github.io/EquiContact-website/