"Pins 1, 2 & 3" mean nothing. You need to get a wiring diagram for that switch. I can only assume it DOES matter which wires go where.
Do you have a model number of the switch?

Unless your heater has a grounded or polarized male plug on the end of the supply cord, it is unlikely that the neutral is even identified within the heater. White color does not necessarily mean that it is a neutral at that moment of use. And if it is identified inside the heater, it is unlikely that is is switched. Does your heater have two speeds or heat levels?

Yes, it's a grounded plug and yes, it has a low and high speed fan.

Unless you, or someone near to your location, has some simple test equipment and knows how to use it, this situation will require some guessing. My guess is that the most distant connection is the power supply from the cord. The other two are the fan speeds. Try it that way-- if you choose to do so-- and see how it works. If you have the two speeds and they are working in reverse from originally, then unplug the cord and switch the two leads that are nearer each other on the switch. But all the risks involved, if any, are for your account and assumed by no one else.

What does this switch control? Does it only control the power and is there a separate switch for the fan hi/low speed?

As you said it is illuminated, I'm assuming that it only controls the power. You'd then have the switched hots and a neutral for the internal neon(?) light bulb.

Yes, it would matter if this was the case, if you reversed the line and load sides of the switch, as the light would always be "on" even if the switch was off.

Simple enough to just reverse the leads but without knowing more information about what you have, or testing it with a meter, I don't think anyone would advise you to simply try it.

Thanks for the responses.

This illuminated switch, located on the front panel, is only for on/off for the heater. There's a separate switch in the back for fan hi/low speed. It appears the 3-wire pigtail comes from the the fan hi/low switch and into this illuminated switch. It's only illuminated to show that it's "on." Am I right to think this is a switch loop type setup?

"Am I right to think this is a switch loop type setup?"

While you could call it a switch loop, within an appliance it's just point to point wiring.

You have a simple illuminated SPST switch. One side of the switch, the line side, goes to the hot on the incoming AC cord.

The other side, the load side, goes to the heating element and fan speed control switch.

The line side also feeds an internal lamp in the switch and requires a neutral connection in order to illuminate it.

These connections can all be confirmed with an ohm-meter or continuity tester. The switched connections will be a simple open/closed and the lamp, usually neon, will have infinite resistance or if it's incandescent or LED, a much higher resistance.

However, you also need to figure out the wiring now to determine the line side, the switched load side, and the neutral.

You can use the ohm-meter, continuity tester, to read from the polarized plug to identify the incoming hot and neutral - the wide blade is the neutral. Obviously the the remaining wire is the load.