Please explain to me how I'm missing the big picture, Luciano, but all I can see right now are pieces of logical legerdemain in most accounts of the Grandi series.
Take your pair of examples:
(1 - 1) + (1 - 1) + (1 - 1) + (1 - 1) + . . .
1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + . . .
I can't help noticing there are eight 1's in the first, but only seven in the second. So it's not just a matter of "grouping them differently". Adding and subtracting the 1's in accordance with the signs and brackets gives 0 and 1 respectively, but then it's hardly surprising that changing the task results in a different solution. To be consistent the second should be:
1) + (-1 +1) + (-1 + 1) + (-1 + 1) + (-1 + . . .
which does sum to 0.
Similarly with your other example in which you claim all you're doing is adding a zero at the start. I just can't "agree that we haven't changed the sum at all", because once again the quantity of 1's gets simultaneously but silently reduced from eight to seven. In other words, a 1 was sneaked away from the second, or if you like, smuggled into the first series. To be consistent the second should go:
0 + 1 - 1 + 1 - 1 + 1 - 1 + 1 - 1 . . .
Restoring that last "- 1" brings the sum back to 0.
I'm not trying to pour cold water on the fun and fruitfulness of manipulating the various items in the Grandi series. Elsewhere in Plus.Maths I offer it as a model for variable stars.