I think you have two main alternatives:

1) Use a MatrixSymbol object with explicit symmetry in its two levels:

In[1]:= A = MatrixSymbol["A", {n, n}, Symmetric[{1, 2}]]

In[2]:= A + Transpose[A] // TensorReduce
Out[2]= 2 MatrixSymbol["A", {n, n}, Complexes, Symmetric[{1, 2}]]

In[3]:= A - Transpose[A] // TensorReduce
Out[3]= SymbolicZerosArray[{n, n}]

In[4]:= Inverse[A] . Transpose[A] // TensorReduce
Out[4]= MatrixPower[MatrixSymbol["A", {n, n}, Complexes, Symmetric[{1, 2}]], 0]

2) or declare assumptions for a symbol B (I use a separate symbol here to avoid Element resolving to True immediately):

In[5]:= $Assumptions = Element[B, Arrays[{n, n}, Symmetric[{1, 2}]]];

In[6]:= B + Transpose[B] // TensorReduce
Out[6]= 2 B

In[7]:= B - Transpose[B] // TensorReduce
Out[7]= SymbolicZerosArray[{n, n}]

In[8]:= Inverse[B] . Transpose[B] // TensorReduce
Out[8]= MatrixPower[B, 0]

Though correct, MatrixPower[squarematrix, 0] should be converted into IdentityMatrix[n].

I think this type of computation requires guidance, mainly to specify what needs to be replaced by what. For example, I'd do something like this:

$Assumptions = {
   Element[A, Matrices[{n, n}, Symmetric[{1, 2}]]],
   Element[x | y | b | c, Vectors[{n}]]
};

rules = {
   x -> Inverse[A] . b,
   y -> Inverse[A] . c
};

In[3]:= b . y == c . x /. rules // TensorReduce
Out[3]= True

Note that the traditional notation that transposes the vector u on the LHS of a two-vector inner product uT . v can be represented more easily in WL as Dot[u, v].

Thank you for your suggestions. Simplification of symbolic array expressions is currently being greatly improved for version WL 14.2. We will explore the type of computation you propose.