If you're trying to switch the pin between a HIGH and LOW state then it's the pin that needs to go to the Common connection, whether that is centre pin of the switch or not. Switches can be different layouts. For example a normal domestic 2 way switch, that you find on your wall for lighting, will have Common at the bottom and then L1 and L2 being your 2 Way contacts at the top.
Common will be the one pin that connects to each of the others alternately when you flick the switch.
IN your case I'd be inclined to use a pullup resistor (internal is fine, but I usually add my own), so that the default state of the input is HIGH. When you switch to the contacts that are connected to Ground then your switch goes LOW. With this, you don't have even a fraction of a second where the contacts might be open, plus you actually only need a 1 Way switch; Open or Closed. In the Open state your input is HIGH via the pullup to 5V. When your switch is Closed, the input is grounded to a LOW state, just the same as when you press a momentary tactile switch, but obviously it holds the LOW state until you switch back.