OK, I guess I have to admit that I have read both volumes by Castro, and what's even worse, I think I have understood them

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So here's a brief summary (in the way I perceive it) of Volume I. If there's interest I will post about Volume 2 one of these days.
There are two kinds of steps in tango: open and cross. A cross step, by definition, is a step with disassociation. It's called cross because the foot which makes the step crosses the imaginary line between the feet when they are together immediately before the step. For example, an ocho is a sequence of two cross steps. (Castro considers front cross and back cross steps separately but this is really not needed when describing the basic structure.)
An open step is just a step which is not cross, that is, any step without disassociation, it could be side, forward or back.
When making a step, the partners can be in parallel feet (they step with the same foot) or in cross feet (they step with opposite feet). They can also move in same direction or in opposite directions. (The correct definition of opposite directions is actually a bit tricky, I will describe it below).
So here it goes the Fundamental Theorem of Tango

:
Consider the following three sets ( "O" stands for open step and "C" for cross step):
A={O/O, C/O, C/C}
(Here, for example, O/O means that the leader takes open step and follower open step, and so on. I don't distinguish between C/O and O/C because the situation is completely symmetric.)
B={Parallel feet, Cross feet}
and
C={Same directions, Opposite directions}
Then (the theorem says) choosing one element from set A and one element from set B determines a unique element from C. Also, choosing one elements from set A and one element from set C determines a unique element from set B. It's quite simple, isn't it?
In other words, only the following 6 combinations are possible:
1. O/O, Par. feet, Same dir.
2. O/C, Par. feet, Opp. dir.
3. C/C, Par. feet, Same dir.
4. O/O, Cross feet, Opp. dir.
5. O/C, Cross feet, Same dir.
6. C/C, Cross feet, Opp. dir.
(Any other combination would destroy the embrace, any kind of embrace, in the sense that the partners won't face each other anymore.)
Now in order for this to be really true we have to define "opposite directions" carefully when one of the partners takes open step and the other cross step. Remember that open step can always be taken in different directions, forward, side or back. So, we will say that the partners have moved in opposite directions if the open step of one of the partners CAN be made in such a way so that the partners would have moved in opposite directions.
Finally, some examples. Salida Americana ( both partners take front cross step) is an example combination 3 above. The follower making back ochos while the leader makes side step is an example of combination 5. Combinations 4 and 6 occur when both the leader and the follower are doing giro around an axis between them (what is sometimes called counter giro). And so on, and so on...
Well, you might wonder where the follower's cross fits in the whole scheme. I might post about this after lunch

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