Which operations are allowed on a digitally signed PDF?

We've found an excellent answer to this question on Stack Overflow: Please take a look at the Adobe technical white paper Adobe Acrobat 9 Digital Signatures, Changes and Improvements, especially its section "Allowed and disallowed changes." Here Adobe clarifies the allowed changes (as seen by Acrobat 9 and up) that can be made to a certified or signed document without invalidating the signatures applied to the document.
  • Allowed actions for certified documents
    Certified with no changes allowed
    Allowed
    No changes allowed
    Disallowed
    Digitally signing
    Supplying form field values
    Adding or editing annotations
    Adding form fields
    Changing page content
    Certified with form fill-in and digital signatures allowed
    Allowed
    Supplying form field values
    Digitally signing
    Disallowed
    Adding or editing annotations
    Adding form fields
    Changing page content
    Certified with annotations, form fill-in, and digital signatures, allowed
    Allowed
    Adding or editing annotations
    Supplying form field values
    Digitally signing
    Disallowed
    Adding form fields
    Changing page content
  • Allowed actions for signed but uncertified documents
    Allowed
    Adding signature fields (see Limitations on adding signature fields to signed but uncertified documents)
    Adding or editing annotations
    Supplying form field values
    Digitally signing
    Disallowed
    Adding form fields other than signature fields
    Changing page content
As you see "Changing page content" always is in the "Disallowed" category. As adding a page changes page content — afterwards there is at least a white page where there used to be nothing —, this is not allowed. (Even though not explicitly mentioned here, instantiating page templates most likely also is allowed whenever form fill-ins are allowed as that would conform to the PDF standard, cf. ISO 32000-1 section 12.8.2.2.2. But making use of such page templates would at least require the document to be specially prepared before signing, and your question sounds like the documents are already signed without any such preparation...) Obviously even these allowed changes need to be added in append mode aka incremental updates. Anything else will break the signature for good. For some backgrounds see this answer.