For T, which is an anti-unitary Z symmetry generator
: T = UKUK = UU = U (U) = Φ,
where Φ is a diagonal matrix of phases. As a result,  and , showing that
:U = Φ U Φ.
This means that the entries in Φ are ±1, as a result of which one may have either . This is specific to the anti-unitarity of T. For a unitary operator, such as the parity, any phase is allowed.
Next, take a Hamiltonian invariant under T. Let |a⟩ and T|a⟩ be two quantum states of the same energy. Now, if , then one finds that the states are orthogonal: a result called Kramers' theorem. This implies that if , then there is a twofold degeneracy in the state. This result in non-relativistic quantum mechanics presages the spin statistics theorem of quantum field theory.
Quantum states that give unitary representations of time reversal, i.e., have  , are characterized by a multiplicative quantum number, sometimes called the T-parity.