Does overloading '==' get you '!='?

If I manually overload the == operator for a structure, do I get the != operator for free (presumably defined to be the boolean opposite), or do I have to overload it manually (even if to just return !(this == rhs)?

Edit-The question is not whether or not I CAN overload both operators, but whether I must overload inequality if I've already overloaded the equality operator. Regardless, good answers have been given.

Overloading operator == does not give you operator !=. You have to do it manually and the canonical way is to implement it in terms of operator == as in !(left == right).

The semantics of the operators are not dictated by the standard. You could very well overload operator == to mean equality yet overload operator != to something different like addition or even equality again (not that this is a good practice, in fact it should be discouraged. When in doubt, do as the ints do...).[Refer (1) Below]

On a side note, Boost.Operators can help you provide canonical implementations for operators. There is also std::rel_ops with a canonical implementation for operator !=.

(1) To know more about it read Three basic rules of operator overloading in C++.

No nothing is for free. You pay for what you use in C++(in case of Operator Overloading).
You only get the Operator which you overload nothing more.

Also, It is a good practice that if you overload == operator then you should overload != as well because the users of your class will expect that to be available.

Operator Overloading C++ FAQ should be a good read.

Answering the updated Question:

The question is not whether or not I CAN overload both operators, but whether I must overload inequality if I've already overloaded the equality operator.

NO.
There is no such requirement that you Must overload != If you need to overload ==. However,it is a good practice that you Should overload operators related to each other.

Why is it a good practice?
Think it from the perspective of the user of your class. If the user of your class can use ==(equality criteria) to compare objects of your class, naturally they are going to expect that they should be able to use !=(Non-equality criteria) as well, this stems from the fact that these two operators are related closely and supported for all built-in tyes.

What happens if you disregard the should and not overload != when you overload ==?
If the users of your class use != they will get a compile error.
They would frown a bit about not being provided with != when they are provided with == and they will have to realign their logic to use == instead of the !=.

So you can live with it but be ready to expect a few frowns and complaints of inconvinience and not providing user friendly interface.