Why can I put a resistor at either lead of this LED and still see its effects?
In short, the brightness of the LED is roughly proportional to the current that flows through it.  The current is determined by the voltage from your source minus the voltage drop across the LED (constant) divided by the total resistance of the circuit (the resistance of your resistor).  The current is what matters for the brightness, and it is the same everywhere in the circuit, and is the same no matter which position the resistor is in.  For more information check out the question linked in the first comment.