After further delving, I found that the iconv() function is included in php5, and this code will (in theory) do the conversion:
// Simple file translation.
$FileToconvert = "fname";
$FileConverted = "fname2";
echo "Converting $FileToconvert ...";
echo "File converted in $FileConverted";
I ran it (in PhpEd) on the site-contents and site-templates SQL dump files. I did a byte-by-byte comparison with Windows Comp and found no differences, and the files are the same size, so apparently, all the characters I have in those files are expressed the same way in both charsets.
The iconv() function does *not* change the default charset and collation statements in the .sql files, but I’m assuming that those would be ignored for a table that already exists, so it seems less likely that charset encoding caused any of my problems.
When I have time, I’ll try to look at whether the conversion from php4 to php5 was an issue. I didn’t use any compatibility mode on either the export or the import that could have been an issue.