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We probed the surface environment of large ($>$80 km in diameter) T-type asteroids, a taxonomic type relatively ill-constrained as an independent group, and discussed their place of origin. We performed spectroscopic observations of two T-type asteroids, (96) Aegle and (570) Kythera, over 2.8--4.0 $μ$m using the Subaru telescope. With other T-types' spectra available in the literature and survey datasets, we strove to find commonalities and global trends in this group. We also utilised the asteroids' polarimetric data and meteorite spectra to constrain their surface texture and composition. Our targets exhibit red $L$-band continuum slopes similar to (1) Ceres and 67P/Churyumov-Gerasimenko, and have an OH-absorption feature with band centres $<$2.8 $μ$m. (96) Aegle hints at a shallow N--H band near 3.1 $μ$m and C--H band of organic materials over 3.4--3.6 $μ$m, whereas no diagnostic bands of water ice and other volatiles exceeding the noise of the data were seen for both asteroids. The large T-type asteroids but (596) Scheila display similar spectral shapes to our targets. $\sim$50 \% of large T-types contain an absorption band near 0.6--0.65 $μ$m likely associated with hydrated minerals. For T-type asteroids (except Jupiter Trojans) of all sizes, we found a weak correlation: the smaller the diameter and the closer the Sun, the redder the visible slope. The 2.9-$μ$m band depths of large T-types suggest that they might have experienced aqueous alteration comparable to Ch-types but more intense than most of the main-belt asteroids. The polarimetric phase curve of the T-types is well described by a particular surface structure and their 0.5--4.0 $μ$m reflectance spectra appear most similar to CI chondrites with grain sizes of $\sim$25--35 $μ$m. Taken as a whole, we propose that large T-type asteroids might be dislodged roughly around 10 au in the early solar system.