I've posted on the forums and it was recommended that I submit a bug report.
When the site name has an apostrophe in it, and I try to use {cms_selflink
dir="start" text="$sitename"}, the text for the link on the page has "'"
where the apostrophe should be.

To recreate the bug:
Install version 1.6.4, and set the site name to "test's site" under Site Admin >
Global Settings. Then create a new page called "test's page". Create another
page after it called "test2's page". In test's page, paste the following code:

{cms_selflink dir="start" text="$sitename"}
{cms_selflink dir="next" text="$title"}
{sitename}
{title}

Everything but the first link works as expected. The first link shows
"test's site" as the text for the link.

It appears that the HTML escape characters are being applied twice. After the
first time, the apostrophe becomes "'" and after the second time, it becomes
"'", which causes it to display as "'" instead of an apostrophe.