Another (similarly overlooked) topic to consider is that when the text Weedpacket quoted says "domain", it means whatever domain for which the cookie was set. The manual explains what the default is for that value, but note that "mysite.com" and "www.mysite.com" are not the same.
So, if a user initially typed in "mysite.com/page1" and you later redirected them to "www.mysite.com/page2", for example, by default any cookies set on the former page load will not be sent for the latter.