The homotopy symmetric $C^*$-algebras are those separable $C*$-algebras for which one can unsuspend in E-theory. We introduce a new simple condition that characterizes homotopy symmetric nuclear $C*$-algebras and use it to show that the property of being homotopy symmetric passes to nuclear $C*$-subalgebras and it has a number of other significant permanence properties. We shall explain that the augmentation ideal of any countable torsion free nilpotent group satisfies this property and discuss a general conjecture for amenable groups. This is joint work with Ulrich Pennig.