  We characterize the operational capabilities of quantum channels which can
neither create nor detect quantum coherence vis-\`a-vis efficiently
manipulating coherence as a resource. We study the class of dephasing-covariant
operations (DIO), unable to detect the coherence of any input state, as well as
introduce an operationally-motivated class of channels $\rho$-DIO which is
tailored to a specific input state. We first show that pure-state
transformations under DIO are completely governed by majorization, establishing
necessary and sufficient conditions for such transformations and adding to the
list of operational paradigms where majorization plays a central role. We then
show that $\rho$-DIO are strictly more powerful: although they cannot detect
the coherence of the input state $\rho$, the operations $\rho$-DIO can distill
more coherence than DIO. However, the advantage disappears in the task of
coherence dilution as well as generally in the asymptotic limit, where both
sets of operations achieve the same rates in all transformations.
