Yes, the apparent contradiction that each other's clocks are running slower is just a very simple matter of perspective.
Here's an easy explanation from the book I'm currently writing:
1. Thus in relativity when two clocks move relative to each other they both see each other’s clocks running at a slower rate due to time dilation, and thus they both see less time passing on the other’s clock than their own.
2. This is easy to understand as a matter of perspective. A spatial analogy will make this clear.
3. Imagine two cars each traveling at the same 60 mph but on roads that are angled with respect to each other. Each measures the speed of the other and the distance it seems to travel in terms of a coordinate grid aligned with the road it’s on. Each car travels entirely along the x-axis in its own coordinate system. Thus each driver sees the other car traveling some distance at some velocity along the y-axis.
4. Using the familiar Pythagorean formula this apparent motion along the y-coordinate reduces the other car’s distance traveled along the x-axis by dx’=√ (dx2 – dy2) and thus its apparent velocity along the x-axis.
5. And the same is true from the perspective of the other car since both measure the other car’s motion relative to their own x-axis.
6. Thus both drivers each see the other car traveling at a slower velocity and covering a lesser distance than their own.
7. But importantly both drivers are viewing the same actual reality from the perspective of their respective coordinate systems.
8. The same is true with respect to time dilation and elapsed proper time. Again each of two relatively moving observers each sees the other’s clock ticking slower and covering a lesser distance through time than their own, and for the exact same reason of perspective.
9. Here too everything is going at the same velocity, the speed of light. So if two clocks are traveling with relative spatial velocity, each will see the other clock ticking slower and covering less distance in time.
10. Here again both observers measure motion through spacetime in terms of a coordinate system in which they are traveling entirely through time with no velocity in space relative to themselves. And both see the other traveling with an equal but opposite relative motion along the x-axis.
11. So using the same Pythagorean theorem, each sees the other’s distance through time traveled as as dτ =√ (dt2 – dx2), less than their own as they see some of the other’s spacetime c velocity as being through space along the x-axis.
12. But again it’s important to understand that both observers are viewing the same actual reality just from the perspectives of their different coordinate systems. In particular they are seeing the actual clock readings of the other clock, and seeing the actual tick rates of each other’s clocks, just from their own native perspective. Each sees the other from the perspective of the frame in which it is at rest.
13. Even though observers in different frames may view the spacetime variables of another clock differently how it’s viewed doesn’t affect its actual behavior at all. The actual behavior of everything in spacetime depends entirely on its own path through spacetime.
14. Specifically its actual elapsed proper time depends entirely on how much it deviates from an inertial path. Otherwise all clocks travel exactly the same distance through time at the same c velocity so long as they follow inertial paths in flat spacetime.
15. How clocks are viewed by other clocks doesn’t affect them in the least, however relativity enables us to calculate any clock’s elapsed proper time from any inertial frame.
16. So all views are perspective views of actual events, but we only see part of a moving clock’s passage through time from our perspective, as do all other observers in relative motion to it, who all must view everything entirely from the perspective of their own coordinate system.
Edgar L. Owen