  Let G be a locally compact group, let $\Omega:G\times G\to \mathbb{C}^*$ 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 property under the
twisted convolution $\circledast$ coming from $\Omega$. We find sufficient
conditions under which $(L^\Phi(G),\circledast)$ becomes a Banach algebra or a
Banach $*$-algebra; we call it a {\it twisted Orlicz algebra}. Furthermore, we
study its harmonic analysis properties, such as symmetry, existence of
functional calculus, regularity, and having Wiener property, mostly for the
case 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 variety of cases.
