Preliminaries
The Lorentz transforms in the Theory of Special Relativity (SR) between frames O and O' defined in the usual way are given by
t' = L (t - vx)x' = L (x - vt)
where v is the relative velocity of the origin of O' with respect to the origin of O (along the x-direction), L = (1 - v^2)^(-1/2) > 1, and t is measured in terms of distance so that the velocity of light, c, becomes equal to 1.
Time Dilation
The usual treatment of Time Dilation is as follows. For a clock situated at the origin of O' we have x'=0, so that
x = vt
and
t' = L (1-v^2) t = t/L
Since L > 1, this means that the clock at the origin of O' runs slower by a factor of L compared to the clock in the frame of O which is coincident with it. This has been called Time Dilation. And the usual interpretation is that the clocks in the frame of O' actually tick slower than those in O. This is further taken to mean that O' physically ages slower than O.
What does the origin of O see?
Consider the perspective of the observer located at the origin of O. She sees clocks of O' going past her with speed v in the x-direction. At any given instant t, at the origin of O (i.e. x=0) we get,
t' = Lt
This means that the clocks of O' coincident with the origin of O run faster by exactly the same factor L. (Ultimately, this is to be expected due to the symmetry of the Lorentz transforms and the Principle of Relativity.)
Now, the observer located at the origin of O would conclude that the clocks of O' run faster than her clock. This is at variance with the observer located at x(=vt) in O who, as we have seen earlier, concludes that the clocks of O' run slower in comparison. In fact, different observers located at different points in the frame of O all come to different conclusions about the rate at which clocks in O' tick. Interestingly, there is a locus of events given by
x = t (L - 1) / (Lv)
where the clocks of O and O' agree. This locus lies between the world lines of O and O'.
Nowhere have I seen the argument that the observations of the observer at the origin of O should be used to conclude that the exact opposite of Time Dilation takes place!
What does all this really mean?
Ultimately, this boils down to the fact that frames O and O' don't agree on the assignment of space-time coordinates to space-time events. According to O, the clocks of O' are not synchronized correctly. And according to O', the clocks of O are not synchronized correctly. However, neither O nor O' can argue that the intrinsic rate at which their individual clocks tick vary merely due to the relative velocity between the two frames. This also follows from the Principle of Relativity because neither frame is privileged. In fact, time as a measurement of physical/dynamical/physiological duration has to tick at the same rate in both frames. This is because the Lorentz transforms are kinematical and do not include any dynamical effects that would affect the physiological rate of ageing. At best, the Lorentz transforms should be interpreted as scaling transforms that maintain covariance between two inertially moving frames (i.e. the speed of light, c, remains 1 in both frames).
