Aizatulin

If your geometry is quite symmetric, you can calculate the covariance matrix from the points (which can be used for calculating an oriented bounding box), to get candidates for the plane. Once you have the box (or the eigenvectors of the matrix) you can check each plane, if it is a symmetrize plane. Problem is, if an object is not 100% symmetric you have to define some tolerances.
Here is a modification of an example (calculating an oriented bounding box) which uses the center and the normal of each face to count the number of symmetric points and take the face (normal) with the highest count. This example is more a first start and may fail some situations.

You can also use attribute transfer node to get normals from another geometry. But as juejuezi mentioned, in fact it can be any vector attribute and VEX has many advantages doing this (creating, manipulating etc ...).

Have you tried extrusion mode with point normal?