Touches on a touch monitor aren't creating Touch objects nor incrementing Input.touchCount

I'm developing some code so that an application will work with mouse & keyboard as well as with touch monitors. Currently, holding the left-click button will let the user move the camera, FPS-style. I would like to add gesture controls so that a touch held on the screen would let the user move the camera in a similar fashion.

The first statement in the update function is

if (Input.GetMouseButton(0) == true|| Input.touchCount == 1)

This statement evaluates as true whether I use the mouse or the touch screen, which seemed good at first. Then I realized that the next statement,

if (Input.touchCount == 1)

never triggers. Instead, touches are interpreted as mouse left-clicks. This causes the first statement to succeed on a touch, but not the second. So the current behavior right now is that holding the mouse left-click button & dragging will swivel the camera, and touching & dragging on the screen does nothing. Any ideas on why this is happening and how to fix it?

The closest I found on these forums was http://answers.unity3d.com/questions/1074752/inputtouchcount-always-zero-on-windows-touch-scree.html but that didn't offer any straightforward solution. Any help would be appreciated!