In this representation of surreal numbers and gaps with classes of ordinal numbers, are all gaps represented?

One way to represent the surreal numbers is the sign representation, that is, as function from a (von Neumann) ordinal to the set $\{+,-\}$.

Now given that there are two values, I got the idea to represent this as set of ordinals, where $+$ means the corresponding ordinal is in the set, and $-$ means it is not. Obviously this exact way doesn't work because there would be e.g. no difference between $0$ (empty string, obviously mapped to the empty set) and $-1$ (the string $-$, represented by the empty set because no $+$ is in the string). But that is easily fixed by adding the domain itself to the set. That is, $0$ would be represented by the set $\{0\} = \{\emptyset\}$, and $-1$ would be represented by the set $\{1\} = \{\{\emptyset\}\}$.

Obviously this way you only get sets of ordinals with a maximal element, and indeed it is easy to see that there's a bijection between the sign representations (and thus the surreal numbers) and the sets of ordinals with maximal element. The maximal element gives the birthday (the domain of the sign function), and the intersection of the set with its largest element (or equivalently, the set with the largest element removed) determines the signs.

It is also not hard to show that the order of the surreal numbers can be defined directly from those sets as follows:

Calculate the symmetric difference of the two sets. Obviously if the symmetric difference is empty, the sets (and thus the surreal numbers) are equal. Otherwise the symmetric difference has a well-defined minimal element (because the ordinals are well-ordered). Of the two original sets, the set containing that minimal element corresponds to the larger surreal number.

Now obviously not all sets of ordinal numbers have a maximal element. Therefore a natural question is what the sets without a maximal element correspond to.

It is easy to see that the order defined above can also applied to those sets. Therefore those sets can be located relative to the surreal numbers. For example, the empty set is clearly seen as smaller than any surreal number, and the set $\omega$ (that is, the von Neumann ordinal, not to be confused with the set representing the surreal number $\omega$, which would be represented by the von-Neumann ordinal $\omega+1$) is larger than any finite surreal number (because those all either have a birthday less than $\omega$, or at least one minus sign), but smaller than any infinite surreal number (because all of those begin with an infinite string of $+$, that is, $\omega$ is a subset of their representation, and the minimal element of the symmetric difference is therefore an infinite ordinal and thus not element of $\omega$).

Based on this, I think that those sets may be associated with gaps of the surreal numbers. But obviously they don't cover all gaps; for example the gap above all surreal numbers is clearly not such a set. But I think this might be fixed by considering classes instead of just sets.

Obviously every class of ordinals that has a maximal element is a set, therefore it is equally true that each surreal number corresponds to a class of ordinals with a maximal element. Also, the well-ordering of the ordinal numbers also guarantees that the order defined above also works on proper classes.

Therefore it is obvious that all classes of ordinal numbers represent gaps of the surreal numbers. For example, the gap above all surreal numbers would then obviously be represented by the class of all ordinals.

However there is one question remaining:

Do the classes of ordinals without maximal element cover all gaps of the surreal numbers? Or are there gaps that are not represented by such a class?