  In a toric symplectic manifold, regular fibres of the moment map are
Lagrangian tori which are called toric fibres. We discuss the question which
two toric fibres are equivalent up to a Hamiltonian diffeomorphism of the
ambient space. On the construction side of this question, we introduce a new
method of constructing equivalences of toric fibres by using a symmetric
version of McDuff's probes (see arXiv:0904.1686 and arXiv:1203.1074). On the
other hand, we derive some obstructions to such equivalence by using Chekanov's
classification of product tori together with a lifting trick from toric
geometry. Furthermore, we conjecture that (iterated) symmetric probes yield all
possible equivalences and prove this conjecture for
$\mathbb{C}^n,\mathbb{C}P^2, \mathbb{C} \times S^2, \mathbb{C}^2 \times T^*S^1,
T^*S^1 \times S^2$ and monotone $S^2 \times S^2$.
  This problem is intimately related to determining the Hamiltonian monodromy
group of toric fibres, i.e. determining which automorphisms of the homology of
the toric fibre can be realized by a Hamiltonian diffeomorphism mapping the
toric fibre in question to itself. For the above list of examples, we determine
the Hamiltonian monodromy group for all toric fibres.
