First, you need to have an answer to a previous question: Can you take free actions during attacks of opportunity?
If you cannot, then you definitely cannot attack take an attack of opportunity with a spiked gauntlet when both of your hands are on the polearm: it’s a free action to let go of the polearm.
Even if you can, it’s unclear if the fact that you could let go of the polearm to hit someone with the spiked gauntlet means you threaten that space. One could easily say that, even though you could let go of it once you are provoked, before you are provoked you aren’t performing an action and therefore cannot let go of the polearm and therefore do not threaten anything.
You can also make a balance-based argument, saying that not threatening interior squares is an intentional drawback of reach weapons that should not be so easily thwarted. Spiked chains (which threaten near and far) are special for a reason. And there are concerns about other free actions you might take during an attack of opportunity and the problems those might cause.
Ultimately, getting free actions during attacks of opportunity is not explicitly spelled out in the rules, and requires fairly “optimistic” reading between the lines. It requires assuming an exception exists even though none is explicitly listed—somewhat dubious. And extending that to allowing you to threaten squares based on a free action you might take is possibly more dubious still. And the limitation on reach weapons not threatening near you is definitely real and intended.
But I still allow it. I think the “implicit” exception makes vastly more sense, and that harping on its absence is fairly pedantic. I think that the fact that a polearm and spiked gauntlet requires investing in two separate weapons still gives spiked chains a large advantage. And plenty of other options—armor spikes, unarmed strikes—don’t require letting go of anything and absolutely work. No free action during an attack of opportunity has ever been a problem for my games. Finally, frankly, I think attacks of opportunity are one of the only nice things that martial characters get, so I think it’s good to allow them to actually do so. So I acknowledge that you probably can’t, strictly speaking, but I think the game is better off if you can—and that there’s enough room there to allow it.