If your "trustworthy" means "valid" (and I infer that it does; please correct me if that's not so), then the only reliable method I can suggest is largely (and painstakingly) manual. I don't know a module that can do all you ask.

1. take a look (with your Mark I eyeball) at the rendered version in an IE version contemporary with the version of MS-Word that produced the page
2. extract only the rendered (editorial) content (something that's amenable to automation)
3. manually build valid html (and CSS) to match any aspect of the writer's rendering you consider important/relevant.

Clearly, it would help to know the character of the material you're trying to scrape/repost: if it's mere blather you're trying to record for historical interest, much of the formatting may be irrelevant; if it's a paper with a lot of math that needs to be rendered just as the writer prepared it, IN THE MS-WORD ORIGINAL, you have a quite different challenge.

Older version of MS-Word use enormously verbose CSS constructs with gay abandon (redundancy) and proprietary xml schemas . The html is typically non-compliant, but not -- as you appear (from the penultimate and final sentences of your first para) to believe -- chiefly by virtue of missing elements ("tags") -- with the exception of an utter failure to single- or double-quote attributes.

Now, fixing the quotes problem can be fairly straightforward. Here's a snippet whose sole purpose is to render a blank line between a table and a normal para:

OK, to fix that your hypothetical module need only be able to parse class=.... and insert quotes between the equal sign and the attribute... and similarly between align=" and its attribute, "center" -- or, preferably, read the whole thing, and throw out everything except <c><p></p>

But (oops!), look at the code for just one piece of the table I mentioned:

<tr style='height:31.5pt'>
  <td width=181 valign=top style='width:135.75pt;border:solid windowte
+xt .5pt;
  border-top:none;mso-border-top-alt:solid windowtext .5pt;padding:0in
+ 5.4pt 0in 5.4pt;
  height:31.5pt'>
[download]

w3c compliant browsers ignore tags they don't understand, so this will probably render more-or-less as intended in most modern browsers, even if you do nothing. But using point measure for sizes, rather than ems gives some browsers headaches; analysing this much hoo-hah for what could have been a simple <tr class="x"><td class="y">...</td></tr> (after the initial overhead of reading the style sheet) chews up ticks/CPU cycles while the would-be reader fidgets ... or maybe goes away mad.

All this assumes your reference to vetting content means you don't want to censor, paraphrase or revise the original writer's work; that your goal is solely translation from MS-html to html.
    ... and, a question: In your opinion or experience, what "tags are missing" from your source?

You could try HTML::Tidy, which is a front-end to tidyp (aka HTML Tidy). I doubt it cleans up your CSS, though.

"tidyp will validate your HTML, and output cleaned-up HTML."

Alternately, for Win users, Tidy (not to be confused with www.tidyp.com/tidyp) can, properly configured, delete most of the MS generated style sections... and on something simple such as shown above, it does a pretty decent job. Perhaps that will be adequate.

Just by the way, tidyp remains a *nix only program. The latest version of Tidy is available as a Win32 application at http://www.paehl.com/open_source/?HTML_Tidy_for_Windows (but you'll want the man and quick ref from sourceforge).

The documents in question are documents, e.g. literary passages, sometimes-elaborate math formatting and so on, all of which are being presented as a part of a vocational test-giving application.   So, the formatting (egregious as it sometimes is...) is important.   My problem is that, sometimes it is incomplete.   Therefore, when my template embeds it into a <div> tag, without the proper closing-tags within the embedded text that <div> tag is not seen as enclosing it anymore.

So, thinking about this requirement a little bit more, I guess that I am really most concerned with “DOM structure” matters ... of making sure that the content, whatever it is, gets wedged into the container.   I really don’t want to delve into the guts of that content.   I simply want to keep it inside the box.   Of course I thought about using the <frame> tag, but the deployment is so bandwidth-constrained that the result looks perfectly dreadful.

You’re right about the Word-generated HTML content, tho’ ... it is hideous.   But, it works.   And I basically want to keep it working.