A combination of low temperature scanning tunneling microscopy (STM) and theoretical calculations is used to investigate Au dimers, supported on thin MgO(001) films, whose thickness was chosen such that charge transfer from the Ag substrate to the deposited Au is possible. Au dimers exist not only in an upright geometry--as theoretically predicted to be the most stable configuration--but also as flat lying dimers which populate a manifold of different azimuthal orientations. Apart from the difference in adsorption configurations, these two isomers exhibit rather different electronic structures: while upright dimers are neutral, flat ones are charged.