> Can the ADC handle the negative voltages directly, or am I
> going to have to offset everything by +2.5V to get it to work?

Yes

As Mike Harrison said, there could be a simple solution, or it
might be necessary to use an op amp

Below -0.6V must be kept away from the ADC pins or the internal
diode conducts and may fry the pin. I had a couple of 877s with
exactly that fault ;-) At least. They were in a circuit measuring voltage
across a motor speed resistor and if the motor stalled under certain
conditions the pin would see a -1V spike. The chips latched up, got
very hot and took a short trip to the bin. Nowadays I protect the ADC
(and any pin that might see a -ve V) with a current-limiting R, a little
C, and a Schottky

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