In the Twin Paradox, a travelling twin returns to the inertial frame of the twin that has stayed at home, and their clocks are compared. The readings on the two clocks are different, but still ticking at the same rate.
Since time and space are treated on the same footing in special relativity, I believe there should be a space-like equivalent of the Twin Paradox where rulers rather than clocks are compared.
Underlying the Twin Paradox is the Clock Effect, which says that in a triangle made with future-timelike vectors, say AB,BC,AC, then the inertial trip AC has a longer proper time than the non-inertial trip AB-BC: for elapsed times, AC > AB+BC. This is the "Reverse Triangle inequality".
A spacelike analogue of this would be the ordinary "Triangle inequality" for a triangle with three spacelike vectors, say PQ, QR, PR. The straight path PR is shorter than the piecewise trip PQ-QR: for distances, PR < PQ+QR.
Of course, what makes the Twin Paradox/Clock Effect puzzling is that it conflicts with our everyday common sense notions of time... which could be called the non-"Clock Effect" for a Galilean spacetime... that is, the absoluteness (the path independence) of time. For a triangle with Galilean future-timelike sides, say MN, NP, MP, for elapsed times, MP=MN+NP.
It is simply this: a traveler who takes the "direct" route between two locations will register a smaller distance traveled than one who goes via a third location.
There is indeed a space-like analogue to the twin paradox. As @robphy mentions, it is the triangle inequality; but it is worth looking at this in a bit more detail. Here's a picture to illustrate it (due originally to Ben Rudiak-Gould and from Wikipedia):
Suppose Jack travels directly from A to C, whereas Jill goes from A to B to C. From Jack's perspective, for each meter forward he travels, Jill travels a longer distance. That is, if $\theta$ is the angle between their paths, Jill travels $\frac{1}{\cos\theta}$ meters forward for each meter forward Jack travels (in the figure we have $\cos\theta = \frac{1}{\sqrt{1+v^2}}$).
But this is entirely symmetric -- that is, from Jill's point of view (using coordinates that are perpendicular to Jill's line of travel) it is Jack who travels more distance for each meter Jill moves forward. If you think of them as being in cars, each sees the other as "behind" them, at least during the initial phase of their journey.
There's no contradiction here: they are measuring different things, since each uses lines perpendicular to their own direction of travel.
So why does everyone agree Jill is the one who has traveled further when they reunite? It is because Jill turned -- she's actually using two different sets of perpendicular lines as her measurements, one set for her outbound journey and one for her inbound journey. These sets overlap, and so when she calculates Jack's total distance she has to subtract the overlap.
The usual twin paradox is basically the same, but using Minkowski spacetime instead of ordinary Euclidean space. Time has the "opposite sign" from space, so instead of Jill having a longer total distance elapsed she has a shorter total time eleapsed. Everyone agrees that Jill is the one who turned around, because in order to do this she accelerated (which is physically detectable)
In your own inertial frame, your clock is constantly advancing (more precisely, it assigns different times to different points on your worldline) but your odometer never changes. So if you and I travel different paths from event $A$ to event $B$, our clocks can show different elapsed times (hence the possibility of a "twin paradox"), whereas both of our odomeeters show zero elapsed distance, so the analogous "paradox" does not arise.
You say you want to treat time and space on the same footing, but the difference is that you can spend your whole life in one place, yet you can't spend your whole life at one time.
Yes indeed. The ladder paradox is what you are searching for. It's all about a ladder moving at high velocity into a garage that is smaller than the ladder proper length. For an observer the ladder cannot fit and for another it can fit. The paradox is resolved taking into account that simultaneity as well as lenght is a relative concept and not an absolute one.
