I figured it out - eventually. Not a CuteEditor issue, but a Dreamweaver issue.
By default, Dreamweaver dotnet generated pages include the page directive:
<%@ Page Language="VB" ContentType="text/html" ResponseEncoding="iso-8859-1" %>
and the meta tag:
<meta http-equiv="Content-Type" content="text/html; charset=iso-8859-1" />
The default charset iso-8859-1, which is a web standard, doesn't include the "curly" quotes and apostrophe, so they get dropped. The civilized solution seems to be to use "UTF-8" Unicode encoding.