  The coupled transport of the charge and orbital angular momentum of electrons
is at the heart of orbitronics. Here, we discuss the reciprocal relation
between the direct and inverse orbital Hall effects (OHEs) in thin films. We
argue that the conventional orbital current is ill-defined as it does not
satisfy the reciprocal relation owing to non-conservation of the orbital
angular momentum. We resolve the problem by adopting the definition of the
so-called \emph{proper} orbital current, which is directly related to orbital
accumulation. We prove the reciprocal relation between the \emph{global}
response of orbital and charge currents. However, we show that their
\emph{local} distributions are generally different, especially due to gigantic
contributions at surfaces, which may lead to unintuitive results when charge
and orbital currents are locally measured. We demonstrate our predictions by
first-principles calculations on W(110) and Pt(111) thin films. In W(110), the
direct and inverse OHEs are severely non-reciprocal locally in each layer
although the total responses are exactly reciprocal. Interestingly, the SHEs
are almost reciprocal locally in each layer. On the other hand, in Pt(111),
both OHEs and SHEs are locally non-reciprocal, which we attribute to the
pronounced spin-orbit interaction. We propose that the locally distinct
responses may be used to distinguish the spin and orbital currents in
experiments.
