Popper recognized — but dismissed as unimportant — that every falsification of a conjecture is simultaneously a confirmation of an opposite conjecture, and every conforming instance of a conjecture is a falsification of an opposite conjecture.
People often notice this thing where falsification and confirmation can be two sides of the same coin. For example, Popper refers to the eddington eclipse test as an example of a theory 'surviving
falsification' but most refer to it as a 'confirmation'. So in this instance the meaning is the same. From here, Popper critics generalise that confirmation and falsification are always two sides of the same coin, and therefore Popper wasn't really saying anything, just playing with language.
But there is a genuine difference between the two. Attempts to falsify makes for tough tests, whereas attempts to confirm are only tough tests in the instances where the test is also acting as an attempt to falsify. So for example, finding another black crow is a valueless confirmation to a popperian in part because it could never falsify that some crows aren't black.
Of course, Popper would avoid this talk of confirmation, because of its association to verification/vindication. (as a historical side note, in the first addition of LSD Popper did refer to corroboration as 'degree of confirmation' and had to change it because it was confusing. But even drawing into the end of his career, in Realism and the Aim of Science he guesses that his followers will eventually have to change 'corroboration' too because, though less confusing, it's still too confusing).
One is that falsifications are much rarer in science than searches for confirming instances. Astronomers look for signs of water on Mars. They do not think they are making efforts to falsify the conjecture that Mars never had water.
one has to be careful not to confuse the psychology of the thinker and the process by which knowledge is improved.
There are many reasons why scientists may be looking for confirmation of water on mars, psychologically, but say when mapping out mars some scientists stumbled upon what seemed like a giant lake. Would their work as scientists be done? No. As Popper says, '...support is little or no value unless we consciously adopt a critical attitude and look out for refutations of our theories."
Falsifications can be as fuzzy and elusive as confirmations.
Popper did know that the auxiliary hypothesis problem applied equally to falsification and confirmation, I'm assuming this is the problem Gardner has in mind. His solution was something like: if you suspect an auxiliary hypothesis is responsible for your test giving the wrong result, then test that auxiliary hypothesis independently. (I incidentally don't buy this answer, but it is as far as I know it's the Popper answer).