There has been a surge of interest in recent years in general-purpose `acceleration' methods that take a sequence of vectors converging to the limit of a fixed point iteration, and produce from it a faster converging sequence. A prototype of these methods that attracted much attention recently is the Anderson Acceleration (AA) procedure. The nonlinear Truncated Generalized Conjugate Residual (nlTGCR) algorithm is an alternative to AA, which was designed from a careful adaptation of the Conjugate Residual method for solving linear systems of equations to the nonlinear context. The various links between nlTGCR and inexact Newton, quasi-Newton, and multisecant methods are exploited to build a method that has strong global convergence properties and that can also exploit symmetry when applicable.