The main purpose of this work is to provide new fractional pseudospectral schemes with equivalence for solving fractional differential equations (FDEs). We develop differential and integral fractional pseudospectral schemes, and prove their equivalence from the distinctive perspective of the Caputo fractional Birkhoff interpolation with zero initial condition. We introduce the notion of fractional pseudospectral differentiation/integration matrices in terms of weighted Lagrange interpolating functions, and provide exact, efficient, and stable approaches to computing these matrices via the Lagrange interpolation and the Jacobi--Gauss quadrature. Numerical results on benchmark FDEs demonstrate the performance of the proposed pseudospectral schemes.