  In 1991 H\'ebrard introduced a factorization of words that turned out to be a
powerful tool for the investigation of a word's scattered factors (also known
as (scattered) subwords or subsequences). Based on this, first Karandikar and
Schnoebelen introduced the notion of $k$-richness and later on Barker et al.
the notion of $k$-universality. In 2022 Fleischmann et al. presented a
generalization of the arch factorization by intersecting the arch factorization
of a word and its reverse. While the authors merely used this factorization for
the investigation of shortest absent scattered factors, in this work we
investigate this new $\alpha$-$\beta$-factorization as such. We characterize
the famous Simon congruence of $k$-universal words in terms of $1$-universal
words. Moreover, we apply these results to binary words. In this special case,
we obtain a full characterization of the classes and calculate the index of the
congruence. Lastly, we start investigating the ternary case, present a full
list of possibilities for $\alpha\beta\alpha$-factors, and characterize their
congruence.
