Electronic – Power supplies in series, can you ground the middle leg

Your switching supply output is isolated from the input via it's transformer. It also has a plastic case which should have a double insulted symbol on it somewhere (two squares, one within the other) This means the output is "floating" (i.e. not referenced to earth)

Never tie the mains neutral to earth, they both serve very different purposes. Although they are actually tied together at the house's input (in most house wiring systems), the neutral is for return currents and the protective earth is for detecting fault currents and keeping (metal) equipment cases at a safe potential.

You sometimes see plastic cased switching supplies with an earth lead, which is usually used for the EMI filter and to reference the output to earth. Sometimes tiny leakage currents through the transformer windings (or the EMI Y caps if the earth is left unconnected) can charge up the output (relative to earth) to a reasonably high voltage, enough to give you an annoying (but safe) zap when you touch one of the terminals. The earth can be used to drain the leakage currents.

So you can tie the isolated output to earth if you wish, but just use the L and N connections as directed on the mains side.

Further reading:

Safety Considerations in Power Supply Design
Power Supply Technical Guide - also deals with other aspects unrelated to the question, but very thorough and detailed.
UL1310 - Class 2 Power Units
UL1012 - Power Units Other Than Class 2

Because the supply is floating, and you only made one measurement at once.

Floating means isolated, which means the resistance between the supply and ground is very very high. So high that the 10 Mohm input resistance of your multimeter is almost a short circuit in comparison.

^{simulate this circuit – Schematic created using CircuitLab}

If you were to make both measurements at once, with identical meters, you would find V- at -6V and V+ at +6V with respect to ground, and 12V of course between V- and V+.