Why is LTE either FDD or TDD?
You are mixing two things modulation and multiple access. SC-FDMA has been chosen because it is battery-friendly modulation. SC-FDMA is used in both FDD and TDD modes.

There are several reasons why both FDD and TDD are defined.

Historically, people want to utilize efficiently the precious spectrum. FDD needs two bands while TDD needs only one.

Economically, LTE is a broadband technology, i.e. it is designed for high speed human data connection, and high speed human data connection favors downlink. For example in China, the dominant (and at the beginning of commercial LTE, the only) market of TDD LTE, Youtube video streaming takes a large part. In this case, reserving a band for UL is kind of wasteful. Moreover, LTE was commercialized in the middle of the Great Recession and operators were very skeptical in CAPEX spending.

Protocally, FDD and TDD are very different in term of multiplexing devices (UE), especially in HARQ-ACK reporting, UE scheduling etc. If you look at the procedure standards of LTE, just PHY layer for example, each mode requires very complicated scheduling. In mixing two modes in one cell, the level of complexity is huge for eNB, with no relevant benefit. However, mixing two modes between two or more cells are considerable and has been standardized since Release 11 in the Carrier Aggregation scope.

Technically, TDD supports better channel information (CSI) reporting thanks to the reciprocity of channel. Thus it is favored for beamforming, which requires accurate channel estimation, and a very good candidate for massive MIMO.

Finally, yes you can mix two modes if you want, but the efforts would not worth the benefit you get.