Python operator overloading using bitwise operators, to define the behavior of bitwise operations for custom classes. By implementing special methods such as `__and__`, `__or__`, `__xor__`, `__lshift__`, `__rshift__`, and `__invert__`, you can customize how bitwise operations work with your objects.

```class Number:
def __init__(self, value):
self.value = value

def __and__(self, other):
if isinstance(other, Number):
return Number(self.value & other.value)
else:
raise TypeError("Unsupported operand type for bitwise and.")

def __or__(self, other):
if isinstance(other, Number):
return Number(self.value | other.value)
else:
raise TypeError("Unsupported operand type for bitwise or.")

def __xor__(self, other):
if isinstance(other, Number):
return Number(self.value ^ other.value)
else:
raise TypeError("Unsupported operand type for bitwise xor.")

def __lshift__(self, other):
if isinstance(other, int):
return Number(self.value << other)
else:
raise TypeError("Unsupported operand type for left shift.")

def __rshift__(self, other):
if isinstance(other, int):
return Number(self.value >> other)
else:
raise TypeError("Unsupported operand type for right shift.")

def __invert__(self):
return Number(~self.value)

# Create two Number objects
num1 = Number(5)
num2 = Number(3)

# Perform bitwise AND operation
result_and = num1 & num2
print(result_and.value)  # Output: 1

# Perform bitwise OR operation
result_or = num1 | num2
print(result_or.value)   # Output: 7

# Perform bitwise XOR operation
result_xor = num1 ^ num2
print(result_xor.value)  # Output: 6

# Perform left shift operation
result_lshift = num1 << 2
print(result_lshift.value)  # Output: 20

# Perform right shift operation
result_rshift = num1 >> 1
print(result_rshift.value)  # Output: 2

# Perform bitwise inversion
result_invert = ~num1
print(result_invert.value)  # Output: -6```

In the example above, the `Number` class overloads the bitwise operators `&` (and), `|` (or), `^` (xor), `<<` (left shift), `>>` (right shift), and `~` (invert). The respective dunder methods are implemented to handle these operations for `Number` objects.

By implementing these methods, we can perform bitwise operations between `Number` objects or between a `Number` object and an integer. The behavior of these operations is defined based on the logic provided in the respective dunder methods.

Note that in the example, the `value` attribute of the `Number` class represents the underlying integer value for the bitwise operations.

By overloading the bitwise operators, you can define custom behavior for bitwise operations in your custom classes, allowing for more meaningful and intuitive bitwise calculations. 