I believe that the most common definition of the switch pattern we are talking about, that is: two parallel tracks, four switches (aren't they actually called turnouts?) connected by a crossing. If you separate the switches into pairs allowing for a crossover movement from only one track to the other, you have what we call a single crossover.
I have never heard, prior to reading the Wickedpedia description, of a "scissors" crossover. I suspect that the terms is more European than American.
In either case, there are only two choices for switch alignment for this track arrangement as Rich points out; either both tracks straight or both tracks diverging. Ya can't have one track going straight and the other diverging. You would have the type of an accident that makes news headlines as a "derailment" with trains lying all over the countryside! So, to make control of the switch as simple as possible for the operator, the best way to control the whole "plant" is with one control switch with only those two options: straight or diverging. How you do that mechanically is your own choice.
Interesting and informative discussion here.
Another interesting track arrangement and somewhat similar to a double crossover is the double slip switch. Now, on this device, you can only have one train passing through at a time, for obvious reasons. But, just like the double crossover, you really can have only two possibilities: straight through the crossing or diverging in either direction. However, many of the track manufacturers have their double slip switches set up requiring two switch motors. This, of course, does then require two control switches. It would be a great feature on a double slip switch to have all points move in such a way that they are either set for straight or curved. Again, it would make control of the plant much easier.
Paul Fischer