I did a quick google search at www.google.com on "ice stalagmites" and found a lot about this. There are many theories and much speculation but the one that seems the most likely has something to do with the mechanics of crystallization of water as it turns to ice. Apparently this phenomenon only happens in plastic containers. Details here: http://www.its.caltech.edu/~atomic/snowcrystals/icespikes/icespikes.htm
I took photos today of an inverted icicle that appeared in frozen water in an upturned plastic chair. Your description of its cause being a process of crystallization in a plastic container therefore makes good sense. It was certainly a weird thing to see.