  We discuss Hamiltonian symmetries and invariants for quantum systems driven
by non-Hermitian Hamiltonians. For time-independent Hermitian Hamiltonians, a
unitary or antiunitary transformation $AHA^\dagger$ that leaves the Hamiltonian
$H$ unchanged represents a symmetry of the Hamiltonian, which implies the
commutativity $[H,A]=0$, and a conservation law, namely the invariance of
expectation values of $A$. For non-Hermitian Hamiltonians, $H^\dagger$ comes
into play as a distinct operator that complements $H$ in generalized unitarity
relations. The above description of symmetries has to be extended to include
also $A$-pseudohermiticity relations of the form $AH=H^\dagger A$. A
superoperator formulation of Hamiltonian symmetries is provided and exemplified
for Hamiltonians of a particle moving in one-dimension considering the set of
$A$ operators forming Klein's 4-group: parity, time-reversal,
parity\&time-reversal, and unity. The link between symmetry and conservation
laws is discussed and shown to be more subtle for non-Hermitian than for
Hermitian Hamiltonians.
