  In hydrodynamic and magnetohydrodynamic turbulence, expressions for the
transfer rates rely on integrals over wavenumber triads $(\textbf{k,p,q})$
satisfying $\textbf{k+p+q=0}$. As an example $S_E^{uu}(\textbf{k|p,q})$ denotes
the kinetic energy transfer rate to the mode $\textbf{k}$, from the two other
modes in the triad, $\textbf{p}$ and $\textbf{q}$. As noted by Kraichnan
(1958), in $S_E^{uu}(\textbf{k|p,q})$, what fraction of the energy transferred
to the mode $\textbf{k}$ originated from $\textbf{p}$ and which from
$\textbf{q}$ is unknown . Such an expression is thus incongruent with the
customary description of turbulence in terms of two-scale energy exchange.
Notwithstanding this issue, Dar etal. (2001) further decomposed these transfers
into separate contributions from $\textbf{p}$-to-$\textbf{k}$ and
$\textbf{q}$-to-$\textbf{k}$, thus introducing the concept of mode-to-mode
transfers that they applied to MHD turbulence. Doing so, they had to set aside
additional transfers circulating within each triad, but failed to calculate
them.
  In the present paper we explain how to derive the complete expressions of the
mode-to-mode transfers, including the circulating transfers. We do it for
kinetic energy and kinetic helicity in hydrodynamic turbulence, for kinetic
energy, magnetic energy and magnetic helicity in MHD turbulence. Separating the
contribution of magnetic advection from magnetic stretching, the energy
mode-to-mode transfer rates involving the magnetic field become uniquely
defined, in striking contrast to the hydrodynamic case. The magnetic helicity
mode-to-mode transfer rate is also found to be uniquely defined, contrary to
kinetic helicity in hydrodynamics. We find that shell-to-shell transfer rates
have the same properties as mode-to-mode transfer rates. Finally calculating
the fluxes, we show how the circulating transfers cancel in accordance with
conservation laws.
