Running an accessibility check on a PDF in PDF by Zaweba
Tools, then Accessibility in PDF by Zaweba runs 32 rules on tags, reading order, alternate text and more, with Fix, Explain and an HTML report.

An accessible PDF is one that people using screen readers and other assistive technology can actually use. The text reads in the right order, pictures are described, headings are marked as headings, tables make sense when read aloud. Many organisations are expected to publish documents like this. PDF by Zaweba includes a checker that tells you where a document stands and helps you put it right.
Choose Tools, then Accessibility. The check runs 32 rules and reports which pass and which don't.
What the rules look at
The rules cover the things that most often make a PDF hard to use. Among them are permission (whether the document's security settings let assistive technology read it), tagged (whether the document has the structure screen readers rely on), language (whether the document says what language it is in, so the right voice is used), title (whether it has a proper title), reading order, alternate text for pictures, tables, lists and headings, and more besides.
You don't need to know the rules by heart. The checker explains each one.
Working through the results
Each result comes with actions. Fix tries to put the problem right for you. Skip rule sets aside a rule that doesn't apply to this document. Explain tells you what the rule is about and why it matters. Check again runs the check once more after you've made changes.
A good way to work is from the top down. Some problems cause others. A document with no tags at all will fail many rules that depend on tags, so fixing that first often clears a whole group of failures in one go. Then check again and look at what is left.
Use Explain freely. Accessibility has its own vocabulary, and reading why a rule exists makes the fixes far less mysterious.
Saving a report
Save report writes the results as an HTML page. That's useful for keeping a record that a document was checked, for sending to a colleague who handles the fixes, or for showing to whoever asked for accessible documents in the first place. Run the check again after fixing and save a second report, and you have a before and after.
How it fits with the other accessibility tools
The checker tells you what is wrong. The other tools do the mending. Tag the document automatically adds structure to an untagged file. The Tags window, the reading order tool and the content windows let you adjust that structure by hand. There are also tools for setting alternate text, the document language and the title, and Make accessible, which runs the built-in action and then alternate text and the check in one sequence. A setup assistant guides you through the steps if you'd rather be led.
Where to start with a new document
For a document you've never checked, run the check first just to see. If the document isn't tagged, tag it automatically, then check again. Set the title and language, which are quick. Add alternate text to the pictures. Then look at reading order, tables and lists, which take more judgement. Finish with a final check and a saved report.
Keeping expectations sensible
An automatic check can confirm a lot, but not everything. It can tell that a picture has alternate text, not whether the text describes the picture well. It can tell that headings are tagged, not whether they're the right headings. So treat a clean report as a strong sign, and then read the document's alternate text and headings once yourself.
Like everything else in PDF by Zaweba, the accessibility check runs offline. Documents that are private as well as needing to be accessible never have to leave your PC.