The X and Y tangent thing also stopped me for a bit, but eventually I worked it out. It really helped me to get control over the bend option on the simple deform modifier.
In Jonathan’s example, the plane is on the floor. The plane’s local Z axis points up, and its X and Y axes are “tangent” to the plane ( I prefer to use the term “coplanar”, but let’s stick with tangent). Blender uses both of the tangent axes to calculate the bend. One of these axes (in this example, the Y axis) tells Blender what line to bend around. The other other tangent axis, the X axis, tells Blender the direction of the bend (up or down). So while the X axis is in its default tangent position, Blender doesn’t know what direction to bend in, which is why the plane doesn’t bend.
There are two ways to fix this:
1) Rotate the object’s axes around the Y axis so the X axis points up or down. This method introduces its own complications, which is why I always use the second method.
2) Preferred method: Introduce another object, usually an Empty, which starts off at the same orientation as the plane, ie the Empty’s X and Y axes lie along the plane, and the Z axis is perpendicular. Then rotate the Empty around the Y axis so the X axis points up or down. I then parent the Empty to the plane, though this is optional.
With this understanding, I now like the simple deform modifier.