  Phase gadgets have proved to be an indispensable tool for reasoning about
ZX-diagrams, being used in optimisation and simulation of quantum circuits and
the theory of measurement-based quantum computation. In this paper we study
phase gadgets for qutrits. We present the flexsymmetric variant of the original
qutrit ZX-calculus, which allows for rewriting that is closer in spirit to the
original (qubit) ZX-calculus. In this calculus phase gadgets look as you would
expect, but there are non-trivial differences in their properties. We devise
new qutrit-specific tricks to extend the graphical Fourier theory of qubits,
resulting in a translation between the 'additive' phase gadgets and a
'multiplicative' counterpart we dub phase multipliers.
  This enables us to generalise the qubit notion of multiple-control to qutrits
in two ways. The first type is controlling on a single tritstring, while the
second type applies the gate a number of times equal to the tritwise
multiplication modulo 3 of the control qutrits.We show how both types of
control can be implemented for any qutrit Z or X phase gate, ancilla-free, and
using only Clifford and phase gates. The first requires a polynomial number of
gates and exponentially small phases, while the second requires an exponential
number of gates, but constant sized phases. This is interesting, because such a
construction is not possible in the qubit setting.
  As an application of these results we find a construction for emulating
arbitrary qubit diagonal unitaries, and specifically find an ancilla-free
emulation for the qubit CCZ gate that only requires three single-qutrit
non-Clifford gates, provably lower than the four T gates needed for qubits with
ancilla.
