Squares whose Digits form Consecutive Integers

Theorem
The sequence of integers whose squares have a decimal representation consisting of the concatenation of $2$ consecutive integers, either increasing or decreasing begins:
 * $91, 428, 573, 727, 846, 7810, 9079, 9901, 36 \, 365, 63 \, 636, 326 \, 734, 673 \, 267, 733 \, 674, \ldots$

This sequence can be divided into two subsequences:

Those where the consecutive integers are increasing:
 * $428, 573, 727, 846, 7810, 36 \, 365, 63 \, 636, 326 \, 734, 673 \, 267, \ldots$

Those where the consecutive integers are decreasing:
 * $91, 9079, 9901, 733 \, 674, 999 \, 001, 88 \, 225 \, 295, \ldots$

Proof
We have:

They can be determined by inspection.

Also see

 * Squares whose Digits form Consecutive Increasing Integers
 * Squares whose Digits form Consecutive Decreasing Integers