It is not necessary to specify monotonicity in any case, as the DNL tells you that and more.
DNL doesn't tell you that. DNL of less than one tells you it is monotonic, greater than one doesn't tell you either way.
A DNL more negative than -1 == non-monotonic. If the DNL is asymmetric it should be specified as e.g., +1.5/-0.5.
the problem here is that no one is fully specifying what they mean.
if |DNL| <= 1 then the device is most certainly monotonic. I think this is what Rufus was referring to.
if |DNL| > 1 then it doesn't tell you either way, and you have to look more closely.
^^^ absolute value here
in the case of asymmetries, then as ejeffrey says, it should be specified, as eg. +1.5/-0.5
if the negative DNL < -1 , then it is non-monotonic.
if the negative DNL is >= -1, then it is monotonic.
if the positive DNL > 1, then it is still monotonic, but not-desirable.
So ejeffrey's example is monotonic, but has at least one step that is more than 1 LSB, and this is thus not desirable in a DAC.
a DNL specified as +0.6/-1.2 would be non-monotonic.
a DNL as +0.5/-0.3 is monotonic.
Often a manufacturer will specify the DNL as simply 1 LSB or ±1 LSB.
In the first case you can assume they mean |DNL| = 1 LSB and therefore it is monotonic, and in the second case it's clear that they mean +1/-1 LSB, which is also the same as |DNL| = 1, therefore it is also monotonic.