Is JavaScript's double equals (==) always symmetric?

It's supposed to be symmetric. However, there is an asymmetric case in some versions of IE:

window == document; // true
document == window; // false

In Javascript, == is always symmetric.

The spec says:

NOTE 2 The equality operators maintain the following invariants:

• A != B is equivalent to !(A == B).
• A == B is equivalent to B == A, except in the order of evaluation of A and B.

The answer to your actual question (is the operator symmetric) is yes. The ECMA-262 spec explicitly states:

NOTE 2 The equality operators maintain the following invariants:

• A != B is equivalent to !(A == B).
• A == B is equivalent to B == A, except in the order of evaluation of A and B.