In Lorentzian relativity, the (strict) past and future light cones are totally disjoint, and no transformation within the part of SO(1, 3) connected with the identity can mix these two sets.
This property is so important that this part of SO(1, 3) is sometimes called the "orthochronous" Lorentz group. It's what guarantees that the causal ordering of events in spacetime is Lorentz invariant, and thus "objective". (I'm tempted to bet that this also enforces some sort of existence and uniqueness requirement on Cauchy initial value problems in Lorentzian spacetime, further formalising the intuition behind causality, but don't quote me on that.)
Space does not have such a linear ordering. A rotation continuously connected to the identity in SO(3) can rotate the +x axis into the -x axis. Therefore, there can be no causal ordering of events in space. This also reflects our intuition about causality: my left can be your right, but if a particle's past from my perspective is its future from yours, we will disagree about the order of causality, which would be a problem.
The reason space doesn't have such a linear ordering is because there are more than two space dimensions, which means that you can rotate space. You can't rotate time alone (in 1+3 dimensions) -- you can only rotate it along with space (Lorentz boosts). The rotation group SO(1) is trivial.
If you had more than one time dimension, then you wouldn't really have a spacetime, you'd have a two conjoined spaces of potentially different dimensions, distinguished by their relative sign in the metric. Your "time" dimension is nondeterministic. (Again, probably something here about Cauchy initial value problems, but again, don't quote me on it.)
(Of course, this ordering is ambiguous as to which direction is future and which is past, and so doesn't imply an arrow of time. T symmetry still exists.)