My MagSafe to MagSafe 2 converter becomes extremely hot. Why? Out of nothing, my MagSafe to MagSafe2 converter became extremely hot while charging. So hot, I could not even touch it.

What could cause it?

I've been using it so I can leave multiple old 85W MagSafe chargers at various places, so I plug my new late 2013 15" Retina MacBook Pro without having to bring a charger.

Edit:

After inspecting for discoloured parts, I saw the flat side of the aluminium surface is a bit discoloured. 

This is what I've done after it got hot:


I cleaned everything a couple of times with finger nails and a wooden toothpick (to prevent scratching: electric stuff doesn't like metal fragments).
The connector kept getting hot.
After turning the connector 180 degrees, it kept cold. If I turned the power plug 180 degrees or 360 degrees, it stays cool too.


Turning 180 degrees was suggested below by mbabineau and seems to fix the issue every time this happens.
Actually it should be RT (room temperature) like all the time. 

Check the pins to see if one of them is getting discolored.

The only reson for your converter to get hot is a bad contact creating a arc, that creates heat.

Making sure the plugs sit tight (on both sides) and that none of the contact pins (on both sides) is discolored. Cleaning the pins could not harm.

Ultimately getting a new converter would be a good idea.