We combine the design of two \emph{random number generators}, \emph{Mersenne Twister} and \emph{Xorgens}, to obtain a new class of generators with heavy-weight characteristic polynomials (exceeded only by the {\sc well} generators) and high speed (comparable with the originals). Tables with parameter combinations are included for state sizes ranging from 521 to 44497 bits and each of the word lengths 32, 64, 128. These generators passed all tests of the \emph{TestU01}-package for each 32-bit integer part and each 64-bit derived real part of the output. We determine \emph{dimension gaps} for 32-bit words, neglecting the non-linear tempering, and compare with an alternative experimental linear tempering.