Ever since special relativity we've had this equation that puts time and space on an equal footing:

$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$

But they're obviously not equivalent, because there's a sign difference between space and time.

Question: how does a relative sign difference lead to a situation where time only flows forward and never backward? We can move back and forth in space, so why does the negative sign mean we can't move back and forth in time? It sounds like something I should know, yet I don't - the only thing I can see is, $dt$ could be positive or negative (corresponding to forwards and backwards in time), but after being squared that sign difference disappears so nothing changes.

Related questions: What grounds the difference between space and time?, What is time, does it flow, and if so what defines its direction? However I'm phrasing this question from a relativity viewpoint, not thermodynamics.