  Anyonic interferometry probes the braiding phases of excitations in
topologically ordered matter. This technique is well established for charged
quasiparticles in the fractional quantum Hall effect. We propose to extend it
to neutral anyons, such as Ising anyons in Kitaev magnets and quasiparticles in
other neutral spin liquids. We find that the thermal current through an
interferometer is sensitive to the statistics of tunneling quasiparticles. We
present a systematic investigation of signatures of various Abelian and
non-Abelian topological orders in Fabry-P\'erot and Mach-Zehnder
interferometers. The heat current through a Fabry-P\'erot device is different
for different topological orders and depends on the topological charge inside
the interferometer. A Mach-Zehnder device shows interference in topologically
trivial systems only. For a non-trivial statistics, the heat current reduces to
the sum of the contributions from two constrictions in the interferometer.
Furthermore, we identify another probe of topological order that involves the
scaling of the thermal current through a single tunneling contact at low
temperatures. The current shows a universal temperature dependence, sensitive
to the topological order in the system.
