Just about every new female student I have taught wanted to "spin". The ballroom community generally refers to dance rotation as "turns" not "spins". Turns are complex to write about because there are such a variety of them. There are four-legged turns (turns danced in partnership), two-legged turns (turns danced solo with no connection to the other partner) and what I call three-legged turns (solo turns with a connection to the other partner). There are also "powered" turns and "un-powered" turns.
Turns are basically accomplished through three methods: 1) foot placement, 2) rotation through the muscles of the leg (from the top of the ankle to the top of the hip) and 3) swivel on the ball of the foot. In general, four-legged turns in Standard/Smooth dancing are accomplished through foot placement and rotation of the muscles in the leg. An example would be a Waltz "Open Left Turn". Another example would be the International Style Viennese Waltz "Natural (Right) Turns" and "Reverse (Left) Turns". Of course there are always exceptions. The "Pivot" and "Spin" turns are examples of four-legged turns that are executed with the use of the "swivel". The "Chair" element is an example where the second step for the lady is a "swivel" and the third step for the lady is usually a "rotation through the leg muscles". The Latin/Rhythm dances typically use "swivel" turns. As expected, there are exceptions here as well. For example, the dance Bolero is based on the "Slip Pivot" element that implements "rotation through the leg muscles". The "Back Spot Turn" in American Style Rumba is another example of the use of muscle rotation.
Let's talk about the "powered" and "un-powered" turns. The leader controls the speed and timing of the turns except for the two-legged solo turn (the music controls this one). In the "powered" turn the leader's body rotation initiates the turn (starts the rotation) and the leaders arm/hand encourages the rotation to its conclusion. It is very important that the leader not muscle the turn with the arm. This action is hard to control and could actually cause injury to the lady turning. An example would be an American Style Silver Waltz "Syncopated Under Arm Turn". The American Style Rumba "Under Arm Turn" is an example of an "un-powered" turn. The leaders hand/arm indicates to the lady where the turn is to be executed. The leader's connection to the lady and his weight changes tells the lady how fast to perform the turn.
As seen above, the two, three, and four-legged turns may use the "swivel" method for turning. The "swivel" is accomplished by rotating on the ball of the foot. The dancer gets on the ball of the foot by bending the knee of the supporting leg (this moves the weight off of the heel and on to the ball) and not by rising through the ankle. The supporting leg must remain toned so that all of the rotation happens between the ball of the shoe and the floor. There is no muscle rotation during the "swivel". If you try to "swivel" with muscle rotation, it is like trying to rotate Jell-O. It gets all distorted and out of control. If the "swivel" is powered, the lady must maintain a connection to the leading hand and not rotate away from the power. She uses the connection for power and control. The rotation is always relative to the center of the entity. The four-legged turn is relative to the center of the partnership. The three and two-legged turns are relative to the center of the dancer turning.
In general, you do want to minimize horizontal movement while you are turning. Deliberately traveling while you are trying to turn does not produce good results. This is generally not a problem in Latin/Rhythm dancing because the dances are relatively stationary. The problem is most common in the traveling dances. Viennese Waltz seems to have the worst problem. The illusion of Viennese Waltz is that it is moving continually in the horizontal plane. The reality is that you move on "one", then the partners rotate and change places on "two" and "three". There is continuous movement all of the time but it isn't all in the horizontal plane. Basically you move, rotate move, rotate, etc. Many dancers try to move horizontally on every step, even while they are trying to rotate. You will always be out of control if you try to rotate and move horizontally at the same time. This makes you feel like you are dancing down hill and going faster and faster and getting more and more out of control. Another example of where this problem exists in the American Style Smooth "Syncopated Under Arm Turn". The two step "swivel" turn happens on the syncopation while on rise. You can see that this is a real problem if the dancer tries to travel while executing the turn.
Hopefully this article will give you a little insight into some fundamentals of turning. Understand that this is just the tip of the iceberg as far as turning is concerned. Ballroom rotation is based on the fundamentals of physics and geometry of the two, three, or four-legged turns.