[ultrafilter]

I think your watch is slow and you think mine is slow in the following sense. For each of many events, I record their time t and position x while you, moving relative to me at constant velocity, observe a different time T and position X. Between two events with the same X-value, such as ticks of your pocket watch, the difference in t-times is larger than the difference in T-times by a factor gamma. Conversely, between two events with the same x-position, such as ticks of my pocket watch, the difference in T-times is larger than the difference in t-times by that same factor gamma.

There is no contradiction because events with identical x-positions do not have identical X-positions.

If you already know what a partial derivative is, see the first paragraph of this: http://mathpages.com/rr/s4-07/4-07.htm