Horses for courses
A lot of what people have said here is correct, but also if I may say a bit generalised.
Ultimately you will find what suits you best for climbing.
Yes, you should definitely train in and out of the saddle, and train your body to push bigger gears, aswell as spinning, because ultimately our perfect individual climbing tempo is dictated by our own physiological capabilities. I would say that from experience most "natural" climbers definitely spin more than mash, and definitely stand regularly, and I would certainly refute that Contador stands to let his opponents know he is there. I think he quite clearly varies his riding position, as is touched on above, to allow different muscle groups to recover.
But that doesn't suit everyone. For example, a rider with a larger percentage of fast twitch muscle fibres, but a poor lung capacity and a poor V02 max, will probably find "spinning" more or less completely impossible to sustain, and standing out of the saddle to climb, a recipe for disaster, but that doesn't mean that rider cannot improve his climbing speed. He just has to go about it differently.
Worth reading on this subject is Robert Millar in the tour special issue of Rouleur.
I would say if you want to improve climbing and enjoy it more....practice some fork management (i.e try to lose any unneccessary weight, which is the most common limiter in climbing), work on your basic leg strength (and your lactate threshold) and play about with different cadences as you climb both short and long climbs.
And finally, certainly don't obsess about your heartrate. Your HR is affected by so many potentially irrelevant factors when climbing, and actually exercising in general, that it represents a pretty inneffective method of gaging precisely training efforts.