Let G be a locally compact group, let Omega : G x G -> C* be a 2-cocycle, and let Phi be a Young function. In this paper, we consider the Orlicz space L-Phi (G) and investigate its algebraic properties under the twisted convolution circle star coming from Omega. We find sufficient conditions under which (L-Phi (G), circle star) becomes a Banach algebra or a Banach *-algebra; we then call it a twisted Orlicz algebra. Furthermore, we study its harmonic analysis properties, such as symmetry, existence of functional calculus, regularity, and the Wiener property, mostly when G is a compactly generated group of polynomial growth. We apply our methods to several important classes of polynomial as well as subexponential weights, and demonstrate that our results could be applied to a variety of cases.