The duality between space and time leads to many dual principles. For example, Fermat’s principle says that light travels between two given points along the path of shortest time. The dual to this is that light travels between given points in time along the path of shortest length in space.
But as long as there are only single particles with constant velocities, the dual principle will be just another form of the principle. It’s when multiple particles are averaged or their acceleration is non-zero that the dual may make a difference.
Is there a complete symmetry between space and time? I have entertained the thought but hesitated because that seems to say space and time are the same. The symmetries of translation and rotation are like that: move there or turn and the physics is just the same as before. But it’s not surprising that reflection symmetry is sometimes violated since one cannot move an object into its reflection.
Can space and time be switched by a movement? Distance and time can be measured together. Consider a distance measuring wheel traveling at constant speed; it measures distance and duration. Which is which — which is the independent variable and which is the dependent variable, which goes in the numerator and which goes in the denominator, distance or duration? It’s an arbitrary choice. So yes, space and time are symmetric; they may be exchanged without changing the laws of physics.
Space and time are inversely symmetric. They have opposite signs in the spacetime metric. They have opposite (complementary) Lorentz transformations. The limiting speed of light has opposite meaning: it is the maximum travel distance for a given travel time, and it is the minimum travel time for a given travel distance.
Language and culture make space and time seem more different than they are. Physically, they are very similar, and form a finite symmetry.