Write a Python function `opposite_signs(x, y)` that checks whether two given integers have opposite signs. The function should return `True` if the integers have opposite signs and `False` otherwise.
#### Example Usage
```python [main.nopy]
print(opposite_signs(1, -2)) # Output: True
print(opposite_signs(3, 2)) # Output: False
print(opposite_signs(-10, -10)) # Output: False
print(opposite_signs(-2, 2)) # Output: True
```
#### Constraints
- The function should handle both positive and negative integers.
- Zero is considered to have no sign, so any comparison involving zero should return `False`.