identify the topological type obtained by gluing sides of the hexagon

Question

Identify the topological type obtained by gluing sides of the hexagon as shown in the picture below

Clearly the boundary is encoded by the word $abcb^{-1}a^{-1}c$

I do not understand how the surface is glued together - could you help me, please?

Edit: This answer is a Klein bottle (see answer below)

Similar problem: see Is this octogon topologically equivalent to the Klein Bottle?

Answer 1

You should get the Klein bottle:

By combining arrows $a$ and $b$ into a single red arrow, and arrows $a^{-1}$ and $b^{-1}$ into another red arrow, we get the first figure below. Now we must glue the red sides together, and blue sides together, so that the arrows line up.