I was astonished that I've never heard about this until reading your post!
Any act in Islam needs a clear backup, but in this case scholars have three different opinions on the matter some said it is permissible other not and others even said it is bida'h among those who said it is permissible you may find scholars like al-Kassani in his book Badai' as-Sanai' بدائع الصنائع from the hanafi school of fiqh and al-Mirdawi -Hanbali scholar- in his al-Insaf الإنصاف . Here the statement of al-Kassani (My own translation take it carefully):
ومنها أن الإمام كلما صلى ترويحة قعد بين الترويحتين قدر ترويحة يسبح ، ويهلل ويكبر ، ويصلي على النبي صلى الله عليه وسلم ويدعو وينتظر أيضا بعد الخامسة قدر ترويحة ; لأنه متوارث من السلف
Among the sunnan of at-Taraweeh is that each time the Imam has prayed a Tarweehah (a set of Taraweeh which ended by tasleem) he sit between two tarweehat for the time of one tarweehah and make tasbeeh (saying subhan Allah), tahleel (Syaing La Ilaha illa Allah) and takbeer (Saying Allahu Akbar) and he sends prayers and blessings to our Prophet () Allahs prayers and blessings be upon him and waits after the fifth for the time of one tarweehah, as this was done so by the salaf.
Ibn al-Humam (Hanafi) in his Fath al-Qadir فتح القدير only confirms the sitting or pausing between two sets of taraweeh as a custom of the people of the two harams (Mekka and Medina). So what people have done in between was up to them and that they make tasbeeh etc. is a possible option. He also made clear that people of Medina used to pray during this 4 optional raka'a's.
Al-Mirdawi said that the view of the madhhab is that du'a after a set of taraweeh is not makrooh, but ibn 'Oqayl -among the hanbalis- considered it as such.
Ibn Hajar al-Haythami (shafi'i) considered sending prayers and blessings to the Prophet () between two sets of taraweeh as bid'ah.
Ibn al-Haaj al-Fasy (maliki) said in his al-Madkhal المدخل (My own translation take it carefully):
فصل في الذكر بعد التسليمتين من صلاة التراويح وينبغي له أن يتجنب ما أحدثوه من الذكر بعد كل تسليمتين من صلاة التراويح، ومن رفع أصواتهم بذلك والمشي على صوت واحد، فإن ذلك كله من البدع وكذلك ينهى عن قول المؤذن بعد ذكرهم بعد التسليمتين من صلاة التراويح: الصلاة يرحمكم الله فإنه محدث أيضاً، والحدث في الدين ممنوع وخير الهدي هدي محمد صلى الله عليه وسلم ثم الخلفاء بعده ثم الصحابة رضوان الله عليهم أجمعين، ولم يذكر عن أحد من السلف فعل ذلك فيسعنا ما وسعهم.
Section about the dhikhr between two taslemaat (two sets) in salat at-taraweeh one should avoid that dhikr what was invovated by some after each tasleem in the taraweeh prayer, and from raising voices with this and going ahead in one voice, as all this is from among the bid'ah also it is not allowed for the Mu'adhin to say after finishing this dhikr: "As-Salat yarhamukum Allah" (The prayer may Allah have mercy with you) as this too is an innovation. And innovation in the religion is forbidden and the best guiadance is that of Muhammad Allah's prayers and blessings be upon him then that of the guided Caliphs then that of the sahaba may Allah be please with all of them. And none has reported from the Salaf that they did so and we should encompass what the have encompassed
So there seem to be a dispute among scholars on the matter and it is best to avoid innovation in the religion.
Also note that taraweeh is only the set of prayers from the moment you say "Allahu Akbar" and start following the imam until the moment you say the tasleem following the imam what you do between two sets of taraweeh is up to you and not included in this. Therfore your taraweeh is certainly valid and correct. The matter of tasbeeh is discussed and could even be a doubtful matter.