Symmetric group has well-known bijection between conjugacy classes and irreducible representations. I.e. both sets are indexed by Young diagrams.

Question: To what extent this bijection is natural ?

Question is somewhat informal, however I hope it can be left in such a form. One of the ways to make it formal is to ask about the properties which distinguish the bijection above among all bijections between the two sets. (E.g. properties like: identity class is mapped to trivial irreps... ).

Background:

It is very standard material that for any finite group the number of conjugacy classes and irreps is the same, however it is striking that it there is no natural bijection between the two sets is known/(and may be not exists at all).

Mathoverflow has already several discussions around the subject,

Bijection between irreducible representations and conjugacy classes of finite groups,

Symmetric group is not the only group with "natural" bijection, some examples are colloected here: Examples of finite groups with “good” bijection(s) between conjugacy classes and irreducible representations?

Related questions: G/[G,G], irreps and conjugacy classes,Duality between conjugacy classes and irreducible characters for finite monoids?, ...