Asymmetric time dilation (moving clock is slow, stationary clock is fast) is an obvious non sequitur (doesn't follow from Einstein's 1905 postulates) but Einstein "derived" it in 1905 because it produced a quantitative (and breathtaking) prediction:

Albert Einstein, On the Electrodynamics of Moving Bodies, 1905: "From this there ensues the following peculiar consequence. If at the points A and B of K there are stationary clocks which, viewed in the stationary system, are synchronous; and if the clock at A is moved with the velocity v along the line AB to B, then on its arrival at B the two clocks no longer synchronize, but the clock moved from A to B lags behind the other which has remained at B by tv^2/2c^2 (up to magnitudes of fourth and higher order), t being the time occupied in the journey from A to B." http://www.fourmilab.ch/etexts/einstein/specrel/www/

Symmetric time dilation (either clock is slow as judged from the other clock's system) is a valid deduction from Einstein's 1905 postulates but is sterile - no prediction at all. If Einstein had obeyed logic and deduced symmetric time dilation in 1905, special relativity would never have been accepted.

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