I didn't think these were the pins you were reading? I thought you were reading 1 and 3?
This behaviour and why this seems to work, has been explained.
Reading those inputs that are grounded, clears the sample capacitor.
OP says however he has to read the ones which are grounded, and then read the ones he actually wants to read.
That suggests he's reading A0, A2, A1, A3 in that order.
If that is actually true, the previous explanation doesn't make sense, but if the order is A0, A1, A2, A3 is read, it absolutely makes sense.
There seems to be nothing to keep OP from trying this by changing the sketch to read A0, A1, A0, A3.
If that also works, OP will know that clearing the capacitor is needed.
Or waiting for it to have discharged on its own.
By the way, i have seen comments in the examples, telling that you have to wait a bit between 2 consecutive analog reads, this thread explained me why.