take one of the ice cubes or broken ice cubes and place in a clear glass. when the ice thaws, take a good look at the water. do you see a cloudy material?
repeat the test using distilled water (may need to purchase) and use the distilled water to clean the ice tray.
several years ago i noticed the same situation. one day, by happenstance, i used distilled water and noticed a difference in how the water froze and the formation of the ice cube - it did not break into several pieces and was not cloudy. upon seeing the ice, i noticed that the ice was much clearer. an ice cube was placed into a clear glass and no cloudy material when the ice thawed - clear water.
when using distilled water to make ice, the water on the top of ice tray will freeze first, and then the interface between water/tray will freeze. water in the middle of ice cube is not quite at maximum volume. therefore, when water encapsulated inside the frozen boundary increases in volume (a change in density), it penetrate the outer boundary, freezes, and strangely forms a needle like stalagmite. the other strange behavior is that the ice cube is solid and does NOT break into many pieces. also, the ice cube is clear.
to this day, i witness this behavior when freezing distilled water using ice trays. if i think about it, i will get photos next time and post. i do not fully agree with Biginch reasoning about circulation; then again, i've been wrong before.
hope this helps.
-pmover