Recover data in Sparsebundle disk Image I'm afraid I'm out of luck but...  I followed the directions in various places, 

# hdiutil -nomount -noverify -noautofsck SecureRecords.sparsebundle
/dev/disk1
# fsck_hfs -drfy /dev/disk1
journal_replay(/dev/disk1) returned 22
** /dev/rdisk1
Using cacheBlockSize=32K cacheTotalBlock=49152 cacheSize=1572864K.
Executing fsck_hfs (version hfs-407.30.1).
Block 2 is not an MDB or Volume Header 
Block 1953163 is not an MDB or Volume Header 
volumeType is 0
0000:  0000 0000 0000 0000 0000 0000 0000 0000       |................|
. . .
01f0:  0000 0000 0000 0000 0000 0000 0000 0000       |................|
unknown volume type 
primary MDB is at block 0 0x00 
alternate MDB is at block 0 0x00 
primary VHB is at block 0 0x00 
alternate VHB is at block 0 0x00 
sector size = 512 0x200 
VolumeObject flags = 0x01 
total sectors for volume = 1953165 0x1dcd8d 
total sectors for embedded volume = 0 0x00 
CheckForClean - unknown volume type 
CheckHFS returned 6, fsmodified = 0


The underlying story is that this was in a directory being synced onto a Google Drive and I'm guessing Google Drive did something.  I last mounted this a month ago, and have used this like this for years.  The dates on the files in the bands/ directory are believable, with a couple bands dated on Feb 18, the last time I did anything.
Well, I'm in luck.  I wandered around google drive, and right-click on the relevant folder showed me activity in March.  And, strangely, 14 of the band files had been deleted. Google drive had an option to restore them, individually.  After that downloading the whole sparse bundle as a .zip file, and expanding it, the bundle is now recognized again.

I think the file deletions happened when I was messing around with google drive settings, changing it from "Ask before deleting everywhere" to "Remove items everywhere", on a computer that was sync-ing a much smaller subset of directories.

Lesson, (a) don't depend on Google for anything (although in fairness Drive did provide the tools to recover), (b) for important things, have multiple backups