Unless a reference value for potential is mentioned, there is no way to calculate the absolute value of the potential at A and B.
The wire joining A and B is ideal, that is, it has no resistance. From the definition of resistance $\Delta V = IR$, the potential difference between points A and B is zero. Hence the potentials at A and B are the same.
As the resistances labelled $R_1$ and $R_3$ are between the same potential difference, they are in parallel. The same argument can be applied to resistances labelled $R_2$ and $R_4$. The two sets of resistances are in series.
What is "symmetric" about this problem, is that the equivalent resistance of both sets of resistances are the same. Hence, the terminal potential difference of the battery ($24V$) is split equally across the two sets ; $12V$ each, which represent the potential difference between each terminal of the battery and the points A and B
Hope this helps.