Systematic solution to my soccer ball puzzle

Solution 1:

My solution (a point outside an edge means a tongue, inside, a groove):

It is not completely symmetric, but still has a lot of symmetry.

The idea of my construction is to fill a half of the ball, then "flip" the filling so that the other half gets filled in inverted manner.

Step 1. Start with the central hexagon of the Schlegel diagram. It takes alternating tongues and grooves.

Step 2. Now we assign the tongues and grooves to the belt between two halves of the ball. This has to be very symmetric, since we are going to flip our construction symmetrically. Each pentagon gets one tongue and one groove in the belt. The hexagons alternate: two tongues, two grooves, two tongues,...

Step 3. Now we fill the inner pentagons symmetrically. Well, almost. I didn't manage to achieve the complete ($60^\circ$-rotational) symmetry, so one of the pentagons near the belt got two grooves instead of three.

Step 4. Fill in hexagons having in mind that they will be inverted to the upper (e.g. only one hexagon with three consecutive tongues and three consecutive grooves is allowed). I did this manually; spent couple of hours trying to get this work. Finally, I decided to spoil the symmetry of pentagons a little, then after several tries found the solution.

Step 5. Invert the construction to the outer half (inverting means replacing tongues with grooves and vice versa).

Animation of the process: