Attaching to the blur event does not work; the onChange event fires first, and kendo's onChange handler will wipe out the input before the onBlur works.
Attaching to the onChange event did work; I used the onChange event to store the user input, and then put the value back into the input... but the kendo onChange handler then fired afterwards, changing it to blank, and then overriding my storage variable when my onChange handler reacted. The jQuery event object was slightly different when from user input instead of kendo event handler and I used the presence of a field to distinguish between the two. This feels like a brittle fix and I hope that the engineers decide to not to clear out the field so that the client-side validator can provide user feedback; in my case it is not a required field so the user may not notice when their date is deleted.
$('#StartDate').change(function (e) {
if (e.bubbles) //This is specific to the user's change event.
startDateInput = ($('#StartDate')).val();
});