  The present work provides a comprehensive study of symmetric-conjugate
operator splitting methods in the context of linear parabolic problems and
demonstrates their additional benefits compared to symmetric splitting methods.
Relevant applications include nonreversible systems and ground state
computations for linear Schr\"odinger equations based on the imaginary time
propagation. Numerical examples confirm the favourable error behaviour of
higher-order symmetric-conjugate splitting methods and illustrate the
usefulness of a time stepsize control, where the local error estimation relies
on the computation of the imaginary parts and thus requires negligible costs.
