Cosets and Normal Sub-group: Relation between the two concepts

We know that for any group, G, a subgroup H is normal-subgroup

**IFF **every right coset of H is also it's left coset i.e. Ha = aH for all 'a' in G

What if there is a group where every right coset is a left coset but with some other element

i.e. Ha = bH where a=/=b (for all the right cosets of H, obviously there is one exception He = eH)

Basically what I am saying is that there is a one-to-one mapping between set of right cosets and set of left cosets.

I am not sure if such a structure in groups is relevant / important / logical.

What does the above tell the relation between bH and aH? Are they same?

Any thoughts??