A symmetric matrix is equal to its transpose.
A multi-vector can be represented as a 2x2 matrices.
But what is the multi-vector equivalent of a symmetric matrix?
For instance
$$ a+xe_0+ye_1+be_0e_1 \cong \pmatrix{a+x & -b+y \\ b+y & a-x} $$
if I pose $b=0$, then the matrix is symmetric. Can I then claim that $a+xe_0+ye_1$ is a symmetric multi vector; does it adhere to the properties of symmetry?
I am especially interested in the 3+1D case (space-time algebra). What is the complete multi-vector that is symmetric in space-time?
You picked the matrix representation $$a +x \mathbf{e}_1 +y \mathbf{e}_1 + b \mathbf{e}_1 \mathbf{e}_2 \sim \begin{bmatrix}a+x & -b+y \\ b+y & a-x\end{bmatrix},$$ but there are lots of possible matrix representations of multivectors. For example, an alternate representation for the 2D multivector basis elements could use the Pauli matrices $$\begin{aligned} 1 &= \begin{bmatrix} 1 & 0 \\ 0 & 1\end{bmatrix} \\ \mathbf{e}_1 &= \begin{bmatrix} 0 & 1 \\ 1 & 0 \\ \end{bmatrix} \\ \mathbf{e}_2 &= \begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix} \\ \mathbf{e}_1 \mathbf{e}_2 &= \begin{bmatrix} 0 & 1 \\ 1 & 0 \\ \end{bmatrix} \begin{bmatrix} 0 & -i \\ i & 0 \\ \end{bmatrix} = \begin{bmatrix} i & 0 \\ 0 & -i\end{bmatrix},\end{aligned}$$ for which you would have $$a + x \mathbf{e}_1 + y \mathbf{e}_1 + b \mathbf{e}_1 \mathbf{e}_2 \sim \begin{bmatrix} a + i b & x - i y \\ x + i y & a - i b\end{bmatrix}.$$ Like your construction, you can also recover all the multivector coordinates from such a representation.
If you construct an matrix representaiton of this sort, the meaning of symmetrical will vary based on the representation chosen. As a general statement, such symmetry is not neccessarily meaningful, although it could be for some specific representations.
If you are looking for matrix representations for the (3,1) space time algebra case, the obvious candidate are the Dirac matrices (of which there are also many representations.)