None of the analogue inputs of this device should ever be negative, and if you are under the impression that input voltage range is -2.5V to +2.5V, you are mistaken.
Safe analogue input potentials on any channel (\$V_{CHn}\$, at pins 8...23) are:
$$ 0V \le V_{CHn} \le V_{CC} $$
In practice this range extends slightly beyond those limits, but I wouldn't recommend going there, since input currents begin to rise when that happens, which will probably ruin conversion accuracy.
The constraint \$-\frac{1}{2}V_{REF} \le V_{IN} \le +\frac{1}{2}V_{REF}\$ is not referring to absolute potentials at any input, it is telling you the maximum and minimum difference between the two potentials presented to the ADC for valid conversion, and those two values will depend on whether you are operating in single-ended or differential mode.
This is summarised here, an extract from page 6 of the datasheet:
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That means, for example, that if you establish these conditions:
- \$V_{COM} = +2.5V\$ (pin 7)
- \$V_{REF-}=0V\$ (pin 30)
- \$V_{REF+}=+5V\$ (pin 29)
- program all channels as single-ended mode
then you have 16 single-ended channels, or 16 potentials \$V_{CH0}\$ to \$V_{CH15}\$ to convert. For each channel, the potentials presented to the ADC for conversion will be:
$$\begin{aligned}V_{IN-} &= V_{COM} = +2.5V \\ \\V_{IN+} &= V_{CHn} \\ \\\end{aligned}$$
The effective ADC input in this scenario, which will be the conversion result, is:
$$\begin{aligned}V_{IN} &= V_{IN+} - V_{IN-} \\ \\&= V_{CHn} - V_{COM} \\ \\&= V_{CHn} - 2.5V\\ \\\end{aligned}$$
As long as \$0V \le V_{CHn} \le +5V\$, the constraint \$-\frac{1}{2}V_{REF} \le V_{IN} \le +\frac{1}{2}V_{REF}\$ is never violated.
I don't know whether that solves your problem, since you now have conversions results going from -2.5 to +2.5 (scaled digitally, of course), for input potentials between 0V and 5V, and conversion accuracy dependent entirely on the quality of your +2.5V source.
You can of course set \$V_{COM} = 0V\$, though, so conversions are with respect to ground, but you still must never present negative potentials to any input, and input potentials over +2.5V will be flagged as over-range. Damage to the device can only occur if \$V_{CHn}\$ strays outside the range 0V to \$V_{CC}\$, so input potentials between +2.5V and \$V_{CC}\$ are safe, but will be out of the ADC's valid conversion range.
In other words, you may set these conditions:
- \$V_{COM} = 0V\$ (pin 7)
- \$V_{REF-}=0V\$ (pin 30)
- \$V_{REF+}=+5V\$ (pin 29)
- program all channels as single-ended mode
and the ADC will convert this value:
$$\begin{aligned}V_{IN} &= V_{IN+} - V_{IN-} \\ \\&= V_{CHn} - V_{COM} \\ \\&= V_{CHn} - 0V \\ \\&= V_{CHn} \\ \\\end{aligned}$$
The conversion will be valid for:
$$ 0V \le V_{CHn} \le +2.5V $$
So, if you adopt this configuration, you may still need to scale your input signals to fit within the range 0V to +2.5V, but no offset would be required.