Yes, I'm replying to myself...

Turns out that 'inbox' (or 'Inbox') isn't one of the blessed folders that can be called by name, you have to dig for the Id and ChangeKey for 'Top of Information Store' and take a look in there to find the folder with a display name of 'Inbox'.

 <code>
$FindFolder->Traversal = 'Shallow';
$FindFolder->FolderShape->BaseShape = 'AllProperties';
$FindFolder->ParentFolderIds->DistinguishedFolderId->Id = 'root';
$result = $client->FindFolder($FindFolder);
$folders = $result->ResponseMessages->FindFolderResponseMessage->RootFolder->Folders->Folder;
foreach ($folders as $folder)
{
    if ('Top of Information Store' == $folder->DisplayName)
    {
        $tois_folder = $folder;
    }
}
$FindFolder = null;
$FindFolder->Traversal = 'Shallow';
$FindFolder->FolderShape->BaseShape = 'AllProperties';
$FindFolder->ParentFolderIds->FolderId->Id = $tois_folder->FolderId->Id;
$FindFolder->ParentFolderIds->FolderId->ChangeKey = $tois_folder->FolderId->ChangeKey;
$result = $client->FindFolder($FindFolder);
$folders = $result->ResponseMessages->FindFolderResponseMessage->RootFolder->Folders->Folder;
foreach ($folders as $folder)
{
     if ('Inbox' == $folder->DisplayName)
     {
          // $folder is now the Inbox
     }
}
</code>