if time is in fact a 4th dimension at right angles to space
I don't think it is. Here's my current conception of it, though I may be wrong:
The time coordinate is at a right angle, but the time direction of your path is only at a right angle to your distance when you're at rest. As you increase your velocity, the angle of the time direction becomes more acute. When you start approaching the speed of light, the time direction approaches being parallel to your space direction, and that's how you get length contraction.
Stand with your arms pointing straight out to your side at 90 degrees and look straight forward. You're going to be moving forward in the direction you're looking, but while your standing still you're arms are pointing in the perpendicular direction. The amount that your arms are pointing forward towards the direction you are looking (traveling) is how much time contributes to your position, which is zero right now at rest. As you begin moving faster in the forward direction, move your arms towards the direction you are facing. This is an increase in the time component of the distance. More of your path includes the time direction the faster you are going. When you reach the speed of light, your arms are pointing straight forward and time component makes up all of your distance (the space component is contracted). A photon doesn't really experience passing through space, its more like the spot it is going to is contracted to its current position.
But all that is just off the cuff so I'm sure I screwed something up. Still, I think that's the jist of it.
The way I understand it, the only way for the time component of your path to be at a true 90 degrees would be if you were not moving. I suppose you could achieve that with the proper inertial frame, but that didn't seem to be what the OP was asking.