This is a variation on the problem Consecutive Numbers . Teachers may need to choose between the two problems - doing both may involve too much overlap.
This problem could replace a "standard practice exercise" for adding and subtracting negative numbers. It provides opportunities for a lot of calculation, in a context of experimenting, conjecturing, testing conjectures, etc.
To direct attention to more than just routine calculation, collate sets of some students' results on the board, and ask the group for general descriptive comments - encouraging conjectures and explanations. Students can then go back to working in pairs to test the validity of what they have heard suggested.
Encourage students to move from "there's always a zero" to the reason why this is true - isolate and examine the cases where it is zero.
At the end of the lesson a plenary discussion can offer students a chance to present their findings, explanations and proofs.
How do you know you have considered all the possible calculations?
Do the answers seem random, or can any/all be predicted?
How do you KNOW that what you say will ALWAYS work?
Students could use Consecutive Numbers for a similar problem which offers opportunities to experiment/conjecture/justify but doesn't require negative numbers.
Students could start by considering the solutions when they add and/or subtract three consecutive negative numbers.
Teachers may like to take a look at the article Adding and Subtracting Negative Numbers
What happens if we allow a $+$ or a $-$ sign before the first number?
What happens if it doesn't have to be FOUR consecutive negatives?