It turns out that any isometry of the euclidian plane to itself must be onto (not just one to one). But there's more!
We can completely categorise all these isometries. It turns out that there are just 4 types, listed below in terms of their degrees of freedom (DOF) -- the number of parameters needed to describe a member of the family:
- reflections about a line -- a 2 DOF family.
- translations along a line (with an additional parameter saying how much to translate by); just 2 DOFs are needed, because a translation is just given by adding a vector.
- glides are a translation and a reflection about the same line.
- rotations around a point (with an additional parameter saying how much to rotate by); a 3 DOF family.
Furthermore, you can represent any translation as two reflections (about 2 parallel lines perpendicular to the line along which you're translating; the distance between the lines is half the translation distance). And you can represent any rotation as 2 reflections (about 2 non-parallel lines through the point around which you're rotating; the angle between the lines is half the rotation). So just the reflections are enough to generate all isometries, and the subgroup of translations and rotations is a normal subgroup of the isometry group.
Note the analogous relationships between translations and rotations, with regard to reflections. This is why translations are sometimes called "infinitesimal rotations" (especially in the latter half of the 19th century).
Many thanks to Noether (not Hilbert's assistant, she wasn't an E2 user, the other one) for pointing out a missing isometry and other erors!