We present a new family of discrete ordinates (Sn) angular quadratures based on discontinuous finite elements (DFEMs) in angle. The angular domain is divided into spherical quadrilaterals (SQs) on the unit sphere surface. Linear discontinuous finite element (LDFE) and quadratic discontinuous finite element (QDFE) basis functions in the direction cosines are defined over each SQ, producing LDFE-SQ and QDFE-SQ angular quadratures, respectively. The new angular quadratures demonstrate more uniform direction and weight distributions than previous DFEM-based angular quadratures, local refinement capability, strictly positive weights, generation to large numbers of directions, and fourth-order accurate high-degree spherical harmonics (SH) integration. Results suggest that particle-conservation errors due to inexact high-degree SH integration rapidly diminish with quadrature refinement and tend to be orders of magnitude smaller than other discretization errors affecting the solution. Results also demonstrate that the performance of the new angular quadratures without local refinement is on par with or better than that of traditional angular quadratures for various radiation transport problems. The performance of the new angular quadratures can be further improved by using local refinement, especially within an adaptive Sn algorithm.