Wiring an On-Off-On-Off pushbutton
The linked page explains it fairly clearly:


  DESCRIPTION
  
  Push once to turn one side on, push again to turn off, push again to turn the other side on, one last push and its off again! It's basically a push-button toggle switch that connects and disconnects the middle pin from either of the two side switches with every click. This clicky switch makes a great power switch or mode toggler. We like this switch because it's breadboard friendly and more fun than a power switch. Can reliably handle up to 500mA and up to 12VDC. 


Your understanding of the sequence is correct but we can summarise it in tabular form:

Sequence table

        C    C  (where C is common pin)
Push    A    B  (where A and B are the other two contacts)
0       0    0
1       1    0
2       0    0
3       0    1
0       0    0
1       1    0 ...



  This is a really satisfying switch.


I can't think of too many useful applications for this. 



Note that by connecting A and B together you can make a simple press to toggle switch.