We're having a problem with our upstairs toilet. It has worked fine for many years, but now we can't get it to flush properly.

Since about two weeks ago, it will usually flush fine on the first try... if it hasn't been used for a long while (perhaps ~12 hours). After that, any flushing attempts will result in the water swirling around the bowl, but not going down.

A couple of times, including right before the time when it started not working, I've noticed a strange gurgling sound immediately after flushing.

We have already tried to remove any blockages with:
-- Plunging it like crazy with a really good plunger
-- Going down the toilet with augor #1, which was borrowed from a plumber friend
-- Doing the same with augor #2, rented from a home-improvement store
-- Using an "air cannon" tool. (Sorry, I don't know the name of it, but it's the thing you pump with your arm to get really high air pressure, put in the toilet over the hole, and then release a blast of air to try to unclog things.)

The upstairs shower and sink, which go to the same plumbing line (sorry, I don't know the term, you can tell I'm not very technical or experienced with this stuff), both drain normally. All the rest of the plumbing in our house is working fine.

We're starting to wonder if maybe it isn't a blockage issue after all... if it was, would it still work fine on the first try after being left alone for a while? Should we be thinking it's some sort of air-venting issue? But if it's a thing with the air vent, why does all the rest of the plumbing in the house work fine?

We're kind of lost and need some ideas... Any help?

Yep. Without air to displace the water, it won't flush.

There may be a separate vent for the toilet. You need to snake the vent from the roof being careful not to loose the auger. Critters make nests there sometimes.

Flushing with a bucket might not work either.

Since you did mention that you get the swirl, I'm hoping your getting the release of water from under the rim. That starts the siphoning action.