How is relativity related to anti-particles?
As far as I know relativity doesn't say anything about antiparticles. But particle physics does. Have a look at the Einstein-de Haas effect which "demonstrates that spin angular momentum is indeed of the same nature as the angular momentum of rotating bodies as conceived in classical mechanics". An electron doesn't rotate like a planet, because as we know from atomic orbitals, electrons "exist as standing waves". And as we also know, they have a Dirac's belt "spinor" nature: "In this sense a Möbius strip is reminiscent of spin ½ particles in quantum mechanics, since such particles must be rotated through two complete rotations in order to be restored to their original state".
GNUFDL spinor depiction by Slawekb, see Wikipedia
Google on positron chirality, and you can read that the positron has the opposite chirality to the electron. IMHO you can appreciate this by making Möbius strips with different chiralities. One with an anticlockwise twist to the left represents the electron. The other with a clockwise twist to the right represents the positron. Or you can play around with something dynamical, such as Adrian Rossiter's torus animations. This gif represents the positron:
Now find a program to reverse the gif, and "play it backwards". If you also flip it horizontally it looks more like your left-handed Möbius strip. So this gif represents the electron:
Please note that the electron and positron aren't actually purple doughnuts. They "exist as standing waves" that look motionless, they have a spherical symmetry, and they don't have any kind of surface. It's quantum field theory, the electron isn't some solid torus or billiard-ball sphere that has a field, field is what it is. This field has spin ½ spinor nature, and these gifs are the best things I can find to get that across.
I have heard that a positron is like an electron moving backward through time.
It isn't. Like Anna said, that's just mathematics. The electron is only a "time-reversed positron" like the gifs above are reversed. Neither are going backwards in time. They just have the opposite chirality. And charge. They have the opposite quantum numbers. All of which cancel each other out in annihilation, which typically results in two gamma photons. For an analogy, if a cyclone met a anticyclone, all you'd be left with is wind. When an electron meets a positron, all you're left with is light.