Possibly this relates to the fact that many inverters are built using N-channel FETs (or previously NPN transistors). At present, many of the most suitable switching devices are of these kinds, which are conveniently used with loads returned to the positive supply.
Other arrangements are possible, e.g. with PMOS transistors, but generally the performance of P-channel is devices is not as good.
There is no inversion of the battery voltage, whatever that means. As Adjuster noted, typically NPN or N-MOSFET transistors are used to alternately switch the two sides of the windings to ground which generates the switching waveform in the transformer. These devices work on a positive voltage, so that is the polarity applied to the transformer center-tap. The applied transistor voltage thus has the same polarity as the center-tap. When the transistor turns on there is a positive voltage applied between the center-tap and ground, which is converted by the transformer action and appears on the output winding(s). There is no voltage inversion in the primary.