You are specifically asking about ROMAN Catholics – correct? Meaning we fall under the Pope and believe he is infallible. We live a life with regard to the Catechism of the Roman Catholic Church?
My answer to you is that there are “Orthodox” Roman Catholic MASSES in select cities which preform the mass in Latin. However, I am not aware of any specific group of Orthodox Roman Catholics that dress or take on a certain style of worship as a “sect” of our church. A seperate “sect” would actually contradict the word “Catholic” – as it means “Universal.”
It really comes down to personal choice as to how conservatively you want to dress, or how seriously you regard the Sabbath day. Our church often reminds us to keep the Sabbath day Holy by NOT working, and trying to refrain from shopping. etc. Same with Advent, Lent, etc. It’s up to you and your family how much, how deeply, how conservatively you want to worship.
Orthodox jews are a different SECT of Judaism by the way they worship. Conservative Judaism, Reform Judaism, Israeli Judaism are all different SECTS of the same religion. Orthodox Jews worship at different levels even if they call themselves “Orthodox.” Most Jews call themselves Orthodox. As found on the internet:
Most of the Jews in the world see themselves as Orthodox Jews, even if they do not keep the Laws of Shabbat and Kashrut 100% as they should, they are still normal traditional Jews. Most of them fast on Yom Kippur, and all of them circumcize their sons, simply because it’s written in the Torah. So why don’t they observe the entire Torah Laws? Not one Jew in the world can observe all of the Laws in the Torah. Some Laws can only be performed in Israel, some Laws cannot be performed at all. Some Laws are too hard for some people, and some people are not aware of all the Laws. But still, they are all equally Orthodox Jews. This is an accepted fact of life: Not all Orthodox Jews, are religious Jews. And no Orthodox Jew is perfect.
Typically one views Orthodox Jews as those who do keep the Shabbat meaning they do NOTHING but pray and do the necessary life functions (meaning no phone calls, no visitors, too) from sun-down Friday to sun-down Saturday. I know Jews who keep the Shabbat but don’t grow beards and wear hats. Some won’t allow television in their homes, other’s live a very forward life.
I hope this makes sense? In Judaism as in Catholicism, how “Orthodox” you live your life is up to you. Organizing a group of Roman Catholics (outside of attending a Latin Mass and enjoying and revering such way of worship) as ORTHODOX is not recognized. When we say ORTHODOX CATHOLIC we are referring to the Greek or Eastern Orthodox Church. They do not fall under the Pope, but can and do receive our Eucharist – as they are brothers and sisters with us in Christ. Their mass is similar – yet different. Even the sign of the cross is done differently.
If you respect your Jewish neighbors and their keeping of Shabbat and what you feel to be a contemplative and prayerful life – yes YOU CAN live your Catholic life in similarity. You can wear conservative clothes. Say your rosary every day. Eat fish or refrain from meat on Fridays. There are all KINDS of rules you can follow to bring yourself closer to God. Even in a more conservative way than you have been – if you are feeling called to do so. There are tons of books and tons of info on the internet regarding prayerful life. But I wouldn’t refer to myself as “Orthodox Catholic” as people will think you are Greek or Eastern Orthodox. There are no “sects” of the Roman Catholic Church – we are what we are. We all worship the same God and say the same readings at the masses – universally (even if our regional prayer styles and worship styles are different, that is culture, NOT sects.). How deeply we worship in our daily life is up to us.
A definition of Orthodox:
Adhering to the accepted or traditional and established faith, especially in religion.
Adhering to the Christian faith as expressed in the early Christian ecumenical creeds.
Of or relating to any of the churches or rites of the Eastern Orthodox Church.
Of or relating to Orthodox Judaism.
Adhering to what is commonly accepted, customary, or traditional: an orthodox view of world affairs.
One that is orthodox.
Orthodox A member of an Eastern Orthodox church.***
Hope that answers your question!