The most elegant solution was to detect only + and - signs that are next to a 
number, and remove all others. For example:

filter_var("ad--td#$@++qsdh-3", FILTER_SANITIZE_NUMBER_INT); // returns -3

Right now, the filter behavior is: 

filter_var("ad--td#$@++qsdh-3", FILTER_SANITIZE_NUMBER_INT); // returns --++-3

Which is VERY bad and HORRIBLY wrong.

The purpose of the sanitisation filters is not to transform data so as to make it valid, it merely "removes undesirable characters" (see http://php.net/manual/en/intro.filter.php ). Though this description is not entirely correct (for instance FILTER_SANITIZE_SPECIAL_CHARS with FILTER_FLAG_ENCODE_HIGH will transform some characters into HTML entities -- in a rather flawed way, I must say, because it arbitrarily assumes a sort of ISO-8859-1 extension), what is clear is that data may very well still be invalid after running the sanitisation filters.

Plus, this matches perfectly the documentation ("Remove all characters except digits, plus and minus sign.")

Yes, it matches the documentation. But filtering "ad--td#$@++qsdh-3" and returning 
"--++-3" is wrong. The filter is not implemented correctly, it should remove all + 
and - characters that are not next to a number. Filtering "ad--td#$@++qsdh-3" 
should return "-3", that's a valid integer.

Nothing says it should return a valid integer, just like the e-mail sanitization filter doesn't have to return a valid e-mail.

In other words, and as stated in the documentation, what you are looking for are 
the validate filters, in this case FILTER_VALIDATE_INT, see 
http://www.php.net/manual/en/filter.filters.validate.php