If the half moon is left one side of the planet, does it face the right on the other side? ?

If you are looking at at the moon in the sky, unless you are on the equator and are looking straight up, you are probably looking at it with your back to the pole. If you are on the southern hemisphere, your back is to the south pole, and if you are on the northern hemisphere, your back is to the north pole. So, if the the current phase of the moon looks like a "C" to someone viewing it from the northern hemisphere (facing south), it will look like a backwards "C" to someone on the southern hemisphere (facing north).