Yikes! I've never seen one like that except for electronic (like z-wave/zigbee) switches.
It's not clear from the post that's the switch - maybe you have more info. I thought it was a simple 3 way switch, where each switch has only 2 positions.
I thought it was a simple set of 3 way switches, too. I presumed "both off positions" meant "both up" and "both down". (This presumes the switches are installed consistently. Flip one of them over and you have 2 different sets of "off positions.")
For many folks, switch positions, and the concept of on and off, only lead to more confusion when dealing with the workings behind three way switching. The easiest way to think of it may be this. You are sending water from the house, through a garden hose, to fill a pool. You decide that you need two parallel hoses running to the pool, but only want to run water through one at a time. You then need a "Y" valve at each end of the parallel hoses so that you can direct the water from one hose to the other. You open the Y valve at the house to send the water through hose "A". It flows to the Y valve at the pool and stops, since that valve is open to hose "B". No worries, you just stay where you are, move the house valve to hose B, and the pool starts to fill. You walk to the pool and want to shut the water off, so you move the valve to hose A, and it's off.
So, inquiring minds ask the question, how is this better than having a single hose with a simple valve at each end? Well, you can now stop and start water flow from each end of the hoses. If you had a single hose set-up, you would often need to walk to the other end of the hose to turn it on, since that's the end that was turned off last.
Electricity works in a somewhat similar manner. If you substitute the hose with a current carrying wire, and the Y valve with a three way switch, you (hopefully) get the concept of three way switching. If you're still confused, don't take it personally. I have spent hours scribbling on scrap cardboard, with a sharpie, while explaining the concept to literally dozens and dozens of people. Everybody from other curious workers on construction sites, to (sadly) more than a few electricians.