In which direction does the positron's magnetic moment point? Suppose I have an electron, with its spin pointing in some direction, so that the magnetic moment caused by the spin is pointing upwards. Then, I hold a positron next to the electron so that their spins are aligned parallel to each other.
In which direction does the positron's magnetic moment point?
(The literature is rather inconsistent when it comes to the sign of the g-factor.)
 A: 
Suppose I have an electron, with its spin pointing in some direction, so that the magnetic moment caused by the spin is pointing upwards.

The magnetic dipole moment of an electron and its spin are always oriented in parallel. The same holds for positrons. Please note, however, that parallel here means that two parallel fingers can point with their tips at each other or with their tips in the same direction.
The difference lays in the phenomenon, that the movement of these subatomic particles in an external magnetic field leads to the deflection into different directions. In the case, the direction of the movement and the direction of the external magnetic field are oriented in a horizontal plane (and not parallel to each other), the electron gets deflected perpendicular to this plane (maybe upwards); then the positron gets deflected into the opposite direction (i.e. downwards).
In short: If the magnetic dipole moment of an electron and a positron are pointing in the same direction, their spins are oriented anti-parallel.
