Can Two Classes Inherit from Each Other at the Same Time?

It was already said in a comment: (public) inheritance models a "is a" relationship.

Forget about C++ for a moment and consider the abstract concept of inheritance and what it models. If A inherits from B then any instance of A is a B. Hence the set of all instances of A is a subset of all instances of B. If all instances of A are instances of B and all instances of B are instances of A, then no inheritance is needed to model the relation between A and B, because they are the exact same type.

Now C++ again. This:

 struct A : B {};
 struct B : A {};

Is not possible for two reasons. First, to make A inherit from B the definition of B must be known, and to make B inherit from A the definition of A must be known. Second, each instance of A contains a subobject of type B and each B contains a subobject of type A and every A .... ad infinitum. It is not possible.