The usual Bergman projection is not globally regular on a general (smoothly bounded) pseudoconvex domain, by a result of Christ. We will discuss a family of "twisted" projections, similar to the Bergman projection, and show they are regular on this class of domains. Because the twist factor defining these projections is related to the boundary defining function of the domains, regularity of the twisted projections implies biholomorphic mappings extend smoothly to the boundary, following the lines of the Bell-Ligocka program.