Let me attempt to answer your question, since your question is about SO(10) GUT model, so I will assume that you have the knowledge of simpler version of GUT namely SU(5) GUT model and also little of group theory.
You have 4 different questions >>>
01. Isn't this term ($\psi^{T} C \psi$) already invariant under SO(10)?
02. Doesn't this term ($\psi^{T} C \psi$) yield Majorana mass terms?
03. Why are terms ($\psi^{T} C \psi \phi_{10}$) of this form invariant ?
04. why does the decomposition quoted above ($16 \times 16= 10+120+126$) tell us which Higgs representation we must use in order to get invariant terms?
Since all these questions are kind of interrelated so while answering sometimes I may not be able to follow the particular ordering but mix things up. I apologize for that. Anyway,
ANS to QUS#01: yes it is! SO(10) is an orthogonal group so any SO(10) invariant quantity (say X) must remain the same under the group rotation, i.e, $O^{T}XO=X$, so you are right about that.
ANS to QUS#02: the term ($\psi^{T} C \psi$) is neither a mass term nor a Yukawa term, it is just an invariant fermion bilinear. To understand how to form fermion bilinears and also how many of such terms possible, please consult any introductory field theory/particle physics book, as for example: Introduction to Elementary Particles--D. Griffiths; chapter#07,page#224,Equation#7.68 [ http://www.amazon.com/Introduction-Elementary-Particles-David-Griffiths/dp/3527406018 ].
ANS to QUS#03 and #04: now, to generate mass for the fermions, at first you need construct Yukawa coupling terms in the Lagrangian. Always remember two things,
(a) structure of Yukawa coupling $\sim$ fermion * fermion * Higgs scalar ,and
(b) Lagrangian always has to be group invariant.
From (a) we get for SO(10), $16\times 16 \times Higgs$ but from (b) this Higgs representation has to be choosen in such a way that such terms are invariant. In this stage you need a little bit of group theory. Group theory says, $16 \times 16= 10+120+126$, which says, fermion * fermion$\neq 1$ ($1=$ group singlet and singlets are group invariants). So, to form Yukawa coupling which has to be group invariant, the only options you have are 10,120 or 126 representations, choosing any other representation under SO(10) will not give you a singlet. To understand in details why other representations of Higgs will not form singlet, please consult with Group Theory for Unified Model Building--R.Slansky; page#105 [http://inspirehep.net/record/10204?ln=en ]. This answers QUS#04 and QUS#03.
Now, while answering QUS#02 above, I was very brief, as to explain it clearly, I need the elements from the last paragraph. So lets go back to your QUS#02.
Now you know that possible Yukawa coupling terms are:
(i) $16_{F} \times 16_{F} \times 10_{H}$
(ii) $16_{F} \times 16_{F} \times 120_{H}$
(iii) $16_{F} \times 16_{F} \times 126_{H}$
,F and H stand for Fermion and Higgs respectively.
For minimality of SO(10) GUT models, 120 representation of Higgs is not used and so I will not talk about it rather concentrate on 10 and 126 dimensional representations.
Lets assume that you already know SU(5) GUT as your question involves SO(10) GUT. Again from group theory one can write down the Branching Rules of the representations that we are interested in for SO(10)-->SU(5) Group Theory for Unified Model Building--R.Slansky; page#106 as:
(A)$16=1+\bar{5}+10$
(B)$10=5+\bar{5}$
(C)$126=1+5+\bar{10}+...$ (dots mean higher dimensional representations that we are not interested in).
Now, if you substitute these in the Yukawa couplings, you will get 10 terms that are invariants, but I will only pick two of them for illustration and to attack your question of Majorana mass that you are interested in.
(i) $1_{F}\times \bar{5}_{F} \times 5_{H}$
(ii) $1_{F}\times 1_{F}\times 1_{H} $
If you know SU(5) GUT you already know that first term (i) is a Dirac mass term where as second term (ii) is Majorana mass term, as their basic forms are:
Dirac mass $\sim$ Right Handed Fermion * Left Handed Fermion * Higgs = R * L * $\langle\phi_{H}\rangle$
Majorana mass $\sim$ Right Handed Fermion * Right Handed Fermion * Higgs = R * R * $\langle\phi_{H}\rangle$
(in $\phi_{H}$ I put $\langle\phi_{H}\rangle$ to represent vacuum expectation value [vev in short], as to give the fermions mass, scalar fields have to get vev [to understand vev, see any introductory field theory book as for example: Quantum Field theory---L.Ryder , Chapter#08] )
because, in (i) $1_{F}$ contains right handed neutrino and $\bar{5}_{F}$ contains left handed neutrino and so Dirac type mass ;on the contrary in (ii) both $1_{F}$ contain right handed neutrino and so Majorana type mass.
Hope that it will help, Thanks!