You can create a symmetrical subdivision surface with or without intersection.
There are two types of symmetry:
Click Symmetry
.
A Tools Palette is displayed:

The following icons inform you about the input selection:
-
surface
-
plane
Select a surface.
Plane Selection
becomes active.
Select the plane.
You can also select the plane first then the surface. In
this case, Surface Selection
becomes active.

The result is the mathematical symmetry by a plane:

The surface is made of two parts separated by the plane. A side of the
surface is automatically chosen, corresponding to the bigger side:

A new Side Selection icon
appears in the Tools Palette to let you choose the side.
Click it or click in the 3D area to invert the side of
the symmetry:

Click to add weight to the
intersection of symmetry. You can also access this command by pressing
ALT+SHIFT+H.
A click on the attraction type icon modifies its state.
The icon has two states,
-
to give a sharp edge:

-
to give a smooth edge:

The value of the weight, displayed on the right of the screen can be
changed between 0 and 100:

Hold the left mouse button and move the mouse to reach
the desired value on the weight slider.
Release the left mouse button when you are satisfied with
the weight added to the surface.
Selecting n number of subdivision surfaces will create n number of subdivision symmetries.