An equivalence class is a subset of the domain. Given a relation $R$ that is reflexive,
symmetric, and transitive, everything in an equivalence class is related through $R$
to everything in the same class.
Another way of putting it is that if you take any element $x$ of the domain, then
the equivalence class of $x$ contains every $y$ such that $xRy.$
Note that because of symmetry and transitivity, $yRx$ as well and $yRz$
whenever $xRz,$ so the equivalence class of $x$ contains every $z$ such that $yRz$
and therefore it is an equivalence class of $y$ as well.
But no element of an equivalence class can relate to an element of any other equivalence
class through $R$; if the two elements were related, they would be in the same class.
If $xRy$ means the bracelet $x$ can be rotated and/or reflected to match $y,$
then you can generate the equivalence class of a bracelet by performing every
possible rotation and/or reflection on it.
You can then take any bracelet that is not in the equivalence class of your first
bracelet, and generate another equivalence class from it.
Since you have only finitely many bracelets, and actually not very many of them,
for this problem you can write out all the equivalence classes by literally listing
every element in each class.