In case this helps anyone later: after much eye-straining forum browsing and rereads of the documentation, I was able to decipher that the correct tag to use is
<tiles:useAttribute id="beanId" name="beanNameHere" classname="theBeanClass" scope="request"/>
.
This declares a Java variable in scope request which can then be accessed like any other bean, in my case using EL, for example:
<c:if test="${beanId.class.simpleName == 'someInterestingClass'}">
<c:out value="${beanId.someInterestingProperty}"/>
</c:if>