When a hockey stick is thrown on ice it simultaneously rotates and translates before stopping. Friction probably plays the main role here, along with the shape of the stick. I think maybe it is due to different forces on friction on either side of the Centre of mass, but I do not know for sure.
There's another component to the solution: nonuniform friction. The blade, even if untaped, has different coatings than does the shaft. The butt end may or may not have a tape roll applied. All this means that, even if you throw the stick with zero spin applied, when it hits the ice it's almost guaranteed that there'll be a nonzero net torque due to the variation in friction along the length of the stick.
And, yes, I am a former hockey player :-) .
BTW, compare this to the standard "bowling ball" problem, which is usually stated as "a bowling ball slides without rolling [down the alley] and at some point rolls without sliding. What's the change in linear velocity' etc. Here, the ball has only a single point in contact with the alley; and even then IRL the ball usually goes thru a period of both rolling and sliding before pure rolling. Have fun solving that Hamiltonian!
It is called conservation laws, conservation of momentum and conservation of angular momentum.
Because friction on the ice is very small, the geometry of the stick is a line with non uniform mass, there will be angular momentum and linear momentum that will be transferred to the ice at the points of contact. To only rotated there should be no linear impulse, which is improbable unless arranged on purpose. The same to only slide, again the initial condition has to be arranged on purpose, not a random fall.