I am going to talk about the "Underarm Turn" that is common to American Style Rumba, Cha Cha, Ballroom Two Step and Samba. It is a three step, un-powered turn. I call this a "To The Rear March" turn. Anyone who has been in the military or a marching band knows the turn I am talking about. The turn starts as a (1) side step followed by a (2) crossing slightly forward step swiveling ½ turn and (3) an in-place turn swiveling ½ turn. The turn is lead on the (1) but the turn actually occurs on the (2) and (3). The U/A Turn, turns to the left or to the right and is fundamentally the same in all four dances. The slight differences in execution of the U/A Turn make them all feel different. This turn exists in other dances as well, but I am only going to describe the four named above.
Rumba is danced to a 4 beat - 3 step timing; "Slow, Quick, Quick"; (2 beats, 1 beat, 1 beat). The U/A Turn is lead on the Slow and executed on the Quick, Quick. The music tempo is slow enough to allow for typical "ball, flat" Latin Motion steps. The body has a complete weight shift on each step. The speed of the turn is controlled by the weight changes of the leader, felt by the follower. This action allows for a very smooth and powerful Rumba turn.
Cha Cha is danced to a 4 beat - 5 step timing; "1, 2, 3, Cha, Cha" or "1, 2, 3, 4, &"; (1 beat, 1 beat, 1 beat, ½ beat, ½ beat). The turn is lead on the "1" and executed on the "2", "3". The Cha Cha turn is faster than Rumba because the tempo of the music is a little faster and there are no slows and there are 5 steps to a measure instead of 3. The speed of the turn is such that the Latin Motion steps only get a "ball, almost flat" step. This creates a little split weight action of the body on the "2", "3" turning steps. This allows for a faster turn because there is not a complete weight shift to the "2" step.
Ballroom Two Step is danced to a 4 beat - 3 step timing; "Slow, Quick, Quick"; (2 beats, 1 beat, 1 beat). The characteristic of Ballroom Two Step is a rising action on the 2nd step ( 1st quick) and a falling action on the 3rd step (2nd Quick). The U/A Turn is lead on the Slow and executed on the Quick, Quick. The 2nd step of the turn (1st Quick) is a "ball" only step. The body weight does not get completely transferred to the ball step, there is some split weight action here too. This creates a pole vaulting action up on the 2nd step and a falling back action on the 3rd step. This creates the "lilting" feeling of Ballroom Two Step during the U/A Turn.
Samba is danced to a 2 beat - 3 step timing; "1, a, 2"; ( ¾ beat, ¼ beat, 1 beat). The U/A Turn is lead on the "1" and executed on the "a", "2". As you can see by the timing, the dancer has to execute the complete turn on the "a", "2"; in 1¼ beats. While executing the U/A Turn, the dancer's body weight must stay on top of the "1" step, only the dancer's leg/foot (ball only) powers the turn on the "a" beat and the dancer's body turns and remains in-place for the "2" beat. There is not enough time for any weight transfer to the "a" step. It reminds me of an in-place "paddle turn". There is a slight lift-powering of the turn by the leader during the "a" step.
The way these turns are executed gives them all a different feeling. The basic physics rule here is that on this type of turn, the less body weight transfer, the faster and more controlled the turn. Many dancers get in trouble by trying to make the U/A Turn feel the same in all of the above dances. The Under Arm Turn in each of the above dances has it's own unique feeling determined by the execution of the turn.