Well, this is certainly a puzzle!

http://www.dailymail.co.uk/travel/travel_news/article-3580640/Aviation-experts-baffled-British-Airways-A380-lands-Heathrow-SQUARE-tyre.html

Austin (British Leyland) tried that with the Allegro's steering wheel back in the seventies - didn't last long...

 

You can get deflated car tyres with similar mis-shapes even though the construction/compounds are slightly different. 

 

Fortunately that aircraft tyre didn't shred itself into pieces - I would have hoped a FOD inspection was done on the runway though..

Perhaps, through a pinhole or similar size leak, it lost it's pressure during cruise altitude.

 

Then upon descent--with the tire rim still having a good seal against the rim, and only a pinhole leak somewhere--the increase in atmospheric pressure compressed the tire into that square shape?

That sounds like a reasonable explanation. The pilot noticed a low tire pressure warning shortly after take off. Perhaps your hypothetical "slow leak" lost just enough pressure to trigger a warning, then continued to loose pressure during the flight.

 

I note that the tire in the photo isn't actually touching the ground, but undoubtedly did rotate a few times during touchdown, but eventually wound up in its current position.

That sounds like a reasonable explanation. The pilot noticed a low tire pressure warning shortly after take off. Perhaps your hypothetical "slow leak" lost just enough pressure to trigger a warning, then continued to loose pressure during the flight.

 

I note that the tire in the photo isn't actually touching the ground, but undoubtedly did rotate a few times during touchdown, but eventually wound up in its current position.

 

Ahh, good to know and supports my theory   I assumed it probably did rotate a few times during touchdown very well could have influenced it's shape.  If it had reasonable pressure in it during landing--enough to actually maintain a tire contact patch on the runway--I think it probably would have blown.