I still haven't seen an answer elsewhere, so...I figured out how to use VBScript to parse the date field in the layer's display expression, so that what shows up when you identify features will be a string representing the date that is actually chronological when sorted.
In the display expression, check advanced, and use something like this:
Function FindLabel ([UPDATEDATE])
yr = DatePart("yyyy",[UPDATEDATE])
mon = DatePart("m",[UPDATEDATE])
if Len(mon)<2 then
mon = "0" & mon
dy = DatePart("d",[UPDATEDATE])
if Len(dy)<2 then
dy = "0" & dy
hr = DatePart("h",[UPDATEDATE])
if Len(hr)<2 then
hr = "0" & hr
min = DatePart("n",[UPDATEDATE])
if Len(min)<2 then
min = "0" & min
sec = DatePart("s",[UPDATEDATE])
if Len(sec)<2 then
sec = "0" & sec
FindLabel = yr & "-" & mon & "-" & dy & " " & hr & ":" & min & ":" & sec
DatePart() to parse out the year, month, day, hour, minute, and second elements (in my case, from a field called UPDATEDATE), defines them as separate variables
sec, and then constructs a string to constitute the display expression with hyphens and colons in between. To make each element sort correctly, the code uses
Len() to get the length of each element, and if the length is <2, it concatenates "0" in front of the value, so 12 won't be sorted before 3, for example.
Maybe there's a simpler way, but this accomplished what I wanted.