magento show / render one (1) block in two (2) places at the same time

Sorry for the weird title, but I just spent half a day trying to find this and I am sure the answer exists somewhere already, but I can't find it... so I'm asking again with a searchable title that others may find one day.

In a responsive magento theme, I want the topLinks block to show up in both the responsive menu that is only visible in mobile view and the footer. As both of those places are controlled by css and page-width, I need them to render in the template at the same time. I have this in page.xml

<block type="page/html_header" name="header" as="header">
    <... lots of other stuff ...>
    <block type="page/template_links" name="top.links" as="topLinks"/>
    <... lots of other stuff ...>
</block>

This works just fine when I call <?php echo $this->getChildHtml('topLinks'); ?>, but when I put the following, also in page.xml only the footer renders and the header does not.

<block type="page/html_footer" name="footer" as="footer" template="page/html/footer.phtml">
    <... lots of other stuff ...>
    <block type="page/template_links" name="top.links" as="topLinks"/>
    <... lots of other stuff ...> -->
</block>

What am I doing wrong, or how do I clone the results of the one block in the second location?