Just an alternate idea I saw: If you dont need high accuracy, you can use a voltage divider to bias -1 to +1 into 0 to 1, and set the analog reference to 1.2, and get almost the same result but skip the op amp.
A voltage divider with 2 equal resistors will give you the voltage halfway between its two inputs. If the bottom of the voltage divider is at 1v, and the top input is at -1, you get 0 out.
If the bottom is at +1, you get 1 out.
I saw a really clever design using 2 1 megaohm resistors, and actually using the aref pin as an output to provide the bias for the divider, setting the reference to the 1.2v setting, putting the signal in the top of the divider and ground and connecting the adc to the output of the divider.
Aref wasnt meant as a current source, but they use 1meg resistors so the current draw will be at most a few microamps.
A filter cap on aref probably draws more than that if vcc is noisy(if there wasnt current to filter we wouldnt need the cap!) So a few uA should be fine.
They put a 1nf capacitor betcan the input and ground to fix the issues with the ADC input current causing voltage drop. It worked up to 100hz. They never tested accuracy because it wasnt needed though. I'd guess that the adc internal reference was the major error source, probably a few percent.
You can also actually bias with a voltage divider to the 5V supply, and get a 0 to ~1.6V output, it's just a little harder to understand.
This simulation explains the concept(5v version)
Left click the output, use view in scope.