I understand that voltage is relative to ground, so that can be negative.
However, I'm currently looking at current-sensors (ACS712 current sensor) and in the performance characteristics table, it specifies the Optimized Accuracy Range.
In the case of this sensor, it's being specified in Amps, ranging from -5 A to +5 A
I can't find anything explaining how you could have a negative amperage. As far as I know, electric current is the rate of flow of electric charge within a (part of a) circuit.
How could the flow of charge sensed by the sensor be negative?
Electric current, in a physical sense, is the rate of flow of electric charge indeed. But charge can flow in one direction or in the opposite direction. That's the reason for positive or negative current: it's a matter of how you set your reference.
Negative current is the flow of charges produced by a negative voltage.
You seem to think that current is the magnitude of the charge flow, like speed is w.r.t change of position. In fact, the current is a vector and it has a direction, like velocity. It's just that in a wire there are only two possible directions for the charges to flow, so the current becomes a scalar with a sign.
Quite simply this IC supplies a positive slope voltage as the output that can be directly translated to a current. If the voltage across pins 1+2 and 3+4 is negative or positive the device will still represent the current as a positive voltage.
The polarity of the voltage is what is significant here. The data sheet implies that if a negative or positive voltage is applied to the aforementioned pins then the IC can handle 5 amps regardless of the polarity.
Let us repeat Oersted’s experiment with a long skinny Flamming valve tube, filament heater at one end and cathode at the other. Then put a much smaller magnetic compass near the middle of the tube.
Warm the filament up. Apply a voltage between the filament and cathode and note the compass needle deflect. Call that a measurement of positive current. Turn the tube off. Call the compass indication zero. Now turn the Fleming tube around and light it up with the same heater and cathode voltages. The compass needle will now defect in the opposite direction from what you previously called positive current. You have just measured negative current (Left hand rule for magnetic flux generated by electrons flowing in a vacuum)
Current flow in a wire is similar, but with a much larger number of electrons pushed in a direction at much tinier average velocity component parallel to the wire. Due to Benjamin Franklin accidentally swapping post-it notes on two Leyden jars (or something like that), electron flow is now called negative current.