Fortunately it's easy to find the devices that poll every 32 seconds. I left tcpdump running overnight, and checked those IPs that had polled my server ~588 times during that time. LaCie's Network Space 2 and 2big NAS devices seem to behave similarly.

Here's another one that is polling my server every 32 seconds, but is unable to set the time. The date seems to be set to 2010-03-18 01:07. Time synchronization is enabled with pool.ntp.org as the time source, but the synchronization status is gray.

14:51:47.154014 IP (tos 0x0, ttl 55, id 2634, offset 0, flags [DF], proto UDP (17), length 76)
    x.x.x.x.37274 > 80.69.172.80.ntp: [udp sum ok] NTPv4, length 48
Client, Leap indicator: (0), Stratum 0 (unspecified), poll 0 (1s), precision 0 Root Delay: 0.000000, Root dispersion: 0.000000, Reference-ID: (unspec)
          Reference Timestamp:  0.000000000
          Originator Timestamp: 0.000000000
          Receive Timestamp:    0.000000000
          Transmit Timestamp:   970765772.631595492 (2066/11/12 01:37:48)
            Originator - Receive Timestamp:  0.000000000
Originator - Transmit Timestamp: 970765772.631595492 (2066/11/12 01:37:48)
        0x0000:  4500 004c 0a4a 4000 3711 d039 b210 ba77  [email protected]
        0x0010:  5045 ac50 919a 007b 0038 123f 2300 0000  PE.P...{.8.?#...
        0x0020:  0000 0000 0000 0000 0000 0000 0000 0000  ................
        0x0030:  0000 0000 0000 0000 0000 0000 0000 0000  ................
        0x0040:  0000 0000 39dc b5cc a1b0 3db2            ....9.....=.
14:51:47.154400 IP (tos 0xc0, ttl 64, id 57721, offset 0, flags [DF], proto UDP (17), length 76) 80.69.172.80.ntp > x.x.x.x.37274: [bad udp cksum 0x6967 -> 0xf603!] NTPv4, length 48 Server, Leap indicator: (0), Stratum 2 (secondary reference), poll 5 (32s), precision -21 Root Delay: 0.003799, Root dispersion: 0.021759, Reference-ID: 194.100.2.198
          Reference Timestamp:  3623921112.850508689 (2014/11/02 14:45:12)
          Originator Timestamp: 970765772.631595492 (2066/11/12 01:37:48)
          Receive Timestamp:    3623921507.154014408 (2014/11/02 14:51:47)
          Transmit Timestamp:   3623921507.154391720 (2014/11/02 14:51:47)
            Originator - Receive Timestamp:  -1641811561.477581053
            Originator - Transmit Timestamp: -1641811561.477203756
        0x0000:  45c0 004c e179 4000 4011 ef49 5045 ac50  E..L.y@[email protected]
        0x0010:  b210 ba77 007b 919a 0038 6967 2402 05eb  ...w.{...8ig$...
        0x0020:  0000 00f9 0000 0592 c264 02c6 d800 a5d8  .........d......
        0x0030:  d9ba eff6 39dc b5cc a1b0 3db2 d800 a763  ....9.....=....c
        0x0040:  276d 7d09 d800 a763 2786 3747            'm}....c'.7G

I don't see 'RATE' in the output, which would signify KoD packets. But that's alright -- it is polling at 32 second intervals, which is what I've specified as the minimum allowed interval.

On the plus side: I changed the minimum interval temporarily to 64 seconds to trigger a KoD reply ("RATE") to that client. When I restarted ntpd with that change, my NTP server was of course not initially synchronized, so it sent a reply with INIT as the refid. Apparently the client moved on to using some other host, because it didn't send any further queries to my server. The time is still set to 2010-03-18, so moving on to some other NTP server did not help with its time synchronization problems.

My concerns with this are:
1) is it really necessary for a device like that to poll every 32 seconds
2) if it does send ntp queries, it'd better be able to set the clock
3) it probably shouldn't be using pool.ntp.org as a default, but at least use some NTP vendor zone.
4) the ntp server should be user-selectable
5) it would be nice if it re-resolved the NTP server's IP address at periodic intervals
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