I'm sure you have seen photographs of snowflakes up close. You will notice that there are hundreds of small crystals of ice. This is the crystal structure of ice. You don't see ice cubes with a crystal structure because they freeze too fast. The water doesn't have enough time to move into the crystal lattice when you freeze the water. This web site shows how the molecules line up in the crystals.
Yes, some ice is denser than water. If you put pressure on regular ice, and give it time to rearrange, the molecules will move into a new crystal lattice which results in the ice being more dense than water. In the first ice crystal, there are spaces between some of the molecules which is not there in the second crystal structure.
With extreme pressure, you can have frozen water at 100 °C.
I've been having a play with some granulated and some icing suger (I think "icing sugar" is the same as "powdered sugar") and the thing that strikes me is that icing sugar is less free flowing than granulated sugar. I would guess this is the reason for the density difference.
You mention in a comment that the packing fraction for spheres does not depend on the size of the spheres. This is true, but spheres will only get anywhere near the theoretical packing fraction if they can slide over each other freely and rearrange themselves into a close packed array. If the spheres stick together your get a flocculate that will have a much lower packing fraction.
So my suggestion is that in icing sugar the grains have a tendancy to stick together rather than flow freely over each other. I'd guess this is just down to particle size. Assuming the adhesion between grains is a surface phenomenon then the adhesive strength won't increase with grain size, so the increased mass and size of larger grains makes them easier to pull apart mechanically. The adhesion might be due to Van der Waals forces, or it could be due to an adsorbed water layer making the grain surface slightly sticky.
Response to comment:
The relationship between sediment density and flocculation is well known in the colloid science world (I was a colloid scientist in a previous life) and indeed it's used in industrial processes. For example this patent describes using flocculation to stabilise zeolite slurries. Although it covers zeolite grains in water the principle is exactly the same. If the slurry is not flocculated all the zeolite grains settle into a close packed sediment at the bottom of the tanker and you can't get them out. If you make the zeolite grains stick together they for a less dense sediment (just like the icing sugar forms a less dense powder).
In industry at least most colloid scientists work with fluid suspensions, and sugar while technically still a suspension, is a suspension of solids in air. The way to probe the effect of particle adhesion on powder density would be to control the grain-grain adhesion and show that changes the density. However I don't know how you would do that for a system in air. In fluids it's easy because you can adsorb surfactants and polymers onto the grain surfaces.
It would be interesting to see what density powders were formed in vacuum. If an adsorbed water layer is responsible for the stickiness it should be reduced in vacuum so the powder density will increase. Also you could try vibrating the powder. If particle adhesion has caused formation of a less dense aggregate then vibrating the powder should increase the density because it has will separate adhered grains.
Best Answer
Becoming less dense with pressure is an energy-producing quality; mass being conserved, volume must increase as pressure rises for density (mass/volume) to have that response. So, I'd say that fulminate of mercury does fit the description; it explodes when compressed. It does that once, and only once, of course, and one might not want to be present at such an event.