It is the inverter that generates pure over modified what you are speaking of is a modified wave not pure. Why a Triac does not do well with modified I don't know I have been told by those that do know.

"Modified sign wave input will not damage the system. Rather it tricks the system and keeps the power ON at a low level, when the system intends the power to be off.
The problem with modified sinewave and Square-ish wave power is that the electronic switching devices, will trigger with semi-sharp changes to the sinewave, and turn on at a low level. This condition in effect causes the heater power to remain on at a small level when the system thinks it's turned the output power OFF.
So, with electronically switched proportional systems, temperature creep occurs while under Inverted DC supply power.

On-Off thermostats that use relays, usually work fine on inverted power. The control is bit sloppier, but, adequate for emergencys. Ideally Both would be nice, Tight control 99.9% of the time, sloppier On/Off control when the house power is Out."
Jeff Washburn Helix controls

From Spyder Robotics web site,

"Question: Are the Herpstat's compatibile with UPS units or 12vdc to 120vac power inverters?

The short answer:
Most computer UPS backup systems and 12vdc to 120vac inverters output a modified sine wave. This will not work with any proportional thermostat. They will work with relay based non-proportional thermostats. In order to work with a proportional thermostat the backup system must produce a "True" or "Pure" sine wave. These systems are usually double the price of a standard UPS or power inverter.

The long answer:
Most proportional thermsotats use a Triac to control the AC output. This component gets triggered and turns off on its own once the AC voltage reachs zero volts. Common UPS systems or power inverters using "modified" sinewaves do not recreate the smooth curve sinewave that comes from your home outlet. Instead they spike the voltage 120vac positive, then drop to zero, then spike to 120vac negative. Because of this there is no way to dim the power. Also in my testing the triac typically will not turn off properly and stays stuck ON once its triggered. In order to have a proportional thermostat work properly off a battery powered (or gas generator for that matter) device it has to output a "True" or "Pure" sine wave. These systems recreate the correct smooth arced sinewave just like your home power. Because of the extra components and technology they are usually double the price.

Dion Brewington
Owner, Spyder Robotics "

I can't speak to other generators but I know that as of 2 years ago all Hondas were Pure sine wave inverted. The cost of a modified 100 watt inverter is under 50$ a pure sine wave inverter is over 300$ that is why cheap gas generators usually don't have them.