On the one hand, I have read that Infinite Time Turing Machines (ITTMs) are able to determine whether a given input corresponds to a real that encodes a well-order (assuming that there exists a fixed, ITTM-computable way to encode a well-order of natural numbers into a single real number).
On the other hand, I have read that there exists a notion of reals recognizable by ITTMs, that is, basically, there exist real numbers that satisfy a given set of properties, but ITTMs are not able to determine whether a given real satisfies these properties.
The question is: are ITTMs able to recognize well-orders for arbitrarily large countable ordinals? If not, what is the smallest (countable) ordinal $\alpha$ such that if a real encodes a well-order of order type $\beta \ge \alpha$, there does not exist an ITTM which is able to recognize this fact?
Strictly speaking, this is a little vague. A given real number (encoding a well-order) by definition corresponds to a unique ordinal $\alpha$. I think what you mean to say is: "are ITTMs able to recognize well-orders for arbitrarily large countable ordinals?"
Regarding your question, there are two different notions. The first part below is a bit longer (but I have tried to repeat the same point a few times).
(1) There exists a single ITTM that will always halt with a $\{0,1\}$ when given an arbitrary real number (say on the input tape), essentially as an input. The $0$ would mean that the real number doesn't represent a well-order (on $\mathbb{N}$) with order-type $\alpha$ (for some countable $\alpha$). $1$ would mean that the real number does represent a well-order (on $\mathbb{N}$) with order-type $\alpha$ (for some countable $\alpha$).
So for the question:
Based on what I said in last paragraph, I will reiterate this point again. There exists a single ITTM which will decide the real number as encoding a well-order or not (for arbitrarily large $\alpha$).
If the real number was present on input tape at the beginning of computation, then it doesn't matter how large or small $\alpha$ is. As long as $\alpha$ is countable and the corresponding real number encodes $\alpha$ (in the sense of well-order), then the given ITTM will run for roughly time $\alpha$ and then halt and give a $1$. If the real number doesn't encode a well-order then the ITTM would still halt and give a $0$ as output at some point.
So in that sense, a single ITTM not only "recognizes" (in the sense that it halts for a positive answer) but actually "decides" (in the sense that it halts for both positive and negative answers) for any given real-number whether it encodes a well-order or not.
(2) On the other hand, consider an ITTM starts from blank input. Then there are few different notions.
One is that it halts after sometime with a real as output on its tape. In that case, the real number on output (if it encodes a well-order at all) will encode a well-order with order-type $<\gamma$ (the $\gamma$ being sup of writeable ordinals).
The second notion is that it never halts but the output becomes stable after a certain time (and never changes after that). In that case the real number on output (if it encodes a well-order at all) will encode a well-order with order-type $<\zeta$ (the $\zeta$ being sup of eventually writeable ordinals).