Given the conclusions of special relativity, why do we perceive and interact with time in such a different manner than space?

I'm having a hard time figuring out how to properly phrase this question so bare with me, but I am trying to understand the apparent differences between space and time in the context of special relativity. The theory points towards a fundamental symmetry between the two, and demonstrates this in some very beautiful ways, but I cannot seem to escape the fact that time and space are two very different things. My primary concern is this:

I am able to conceive of (and indeed observe) two events happening at the same time in different spacial locations. I can - in other words - "see" a domain of space, a range if you will. I can see what is in front of me, to the left of me, etc... And yet, I am unable to "see" into the future or the past in such a direct manner... Now I am aware that this is the kind of topic in which intuition and experience can often get in the way and trick me. I am fully prepared to accept that my experiences are distorted and wrong, but is there some physical explanation for why we experience time so differently than space? Why is it that I can move through space in any direction I want but have no control over how I move through time (except implicitly by my motion through space).

Why do we seem to be free to travel about space however we please but all are constrained to follow the arrow of time together? Reading about simultaneity, they speak of events occurring at the same time in different places, or how one event will occur sooner than the other in a different frame of reference. I am totally fine with this. But if time and space are so symmetric, should there not be an equivalent argument for events seperated in time but seeming to occur in different places in different frames?

If space and time are to be treated on the same footing, then why are there such clear differences between the two? Why does time appear only to flow in one direction? Why is there no equaivalent principle of simultaneity for space as there is for time? (or is there?) If the two are equivalent, why are they so different?!

2. Time does not "flow in one direction" because time does not flow. If you think of Minkowski space $M$ as a Lorentzian manifold with a choice of time orientation, and if you reverse that time orientation, you get a different Lorentzian-manifold-with-time-orientation that is isomorphic to $M$. This means, in effect, that the only difference (at least according to SR) between the past and the future is that we've arbitrarily chosen one of them to call "the past" and one to call "the future".