I stumbled across the answer, so I'm posting it here in case anyone else has a similar problem. In my case I had references under the Excel key (1.7 and 1.9).
I backed up the PC, exported the Registry Key, then deleted the 1.9 key.
Word 2010 now opens Excel 2010 spreadsheets beautifully. Thanks Palisade!

THIS ARTICLE IS BASED ON AN ARTICLE FROM PALISADE SOFTWARE

Removing Outdated References to Office from the System Registry
Removing a version of Microsoft Office can sometimes leave behind "orphan" keys in the System Registry. These references to products which are no longer installed can prevent add-ins from working correctly with Excel, You may see messages such as "Application-defined or object-defined error", "Automation error: Library not registered", "Error in loading DLL", "Could not contact the Microsoft Excel application", "File name or class name not found during Automation operation", or "Object variable or with block variable not set
To remove the outdated references, you will need to edit the System Registry, as detailed below
1. Click Start » Run, type REGEDIT and click OK.
2. Click on Computer at the top of the left-hand panel, then press Ctrl+F to bring up the search window.
Paste the following string including the curly braces {...}, into the search window:
(this is the Key for Excel)
{00020813-0000-0000-C000-000000000046}
3. Check (tick) the Keys box and Match whole string only; clear Values and Data.
4. Click the + sign at the left of {00020813-0000-0000-C000-000000000046} to expand it. You will see one or more subkeys:
1.5 for Excel 2003.
1.6 for Excel 2007.
1.7 for Excel 2010.
1.8 for Excel 2013.
1.9 for Excel 2016.
Identify the one(s) that do not match the version(s) of Excel you actually have installed. If all of them do match installed Excel versions, omit steps 5 and 6.
5. You are about to delete the key(s) that correspond to versions of Microsoft Excel that you do not have. For safety's sake, you may want to back them up first. Right-click on {00020813-0000-0000-C000-000000000046}, select Export, and save the file where you'll be able to find it.
6. Right-click the “1.x” key that does not belong, select Delete, and confirm the deletion. Repeat for each 1.something key that does not belong.
7. The {00020813-0000-0000-C000-000000000046} key can occur in more places. Usually they all have the same subkeys, but not always, so you need to examine each instance. Tap the F3 key to get to each of the others in turn. For each one, repeat steps 4 through 6 (click the + sign, export the key to a new file, and delete the orphaned 1.something entries).

Next, repeat the above process using the Key for Office, which is:-
{2DF8D04C-5BFA-101B-BDE5-00AA0044DE52}

The subkeys for Office are
2.3 for Office 2003.
2.4 for Office 2007.
2.5 for Office 2010.
2.6 and 2.7 for Office 2013.
(2.6 and 2.7 are okay for Office 2016 as well, if there is a reference to Office16 under 2.7.)
2.8 for Office 2016.

Your software should now run as normal.