Example when the tail $\sigma$-algebra is not generated by the even and odd tail $\sigma$-algebras 
The even tail $\sigma$-algebra $T^0$ and odd tail $\sigma$-algebra $T^1$ are always contained in the tail $\sigma$-algebra $T$.
  Now I want to prove that it may happen that $T\neq\sigma(T^0,T^1)$.

I have tried defining different families of $\sigma$-algebras (converging vs non converging sequences, starting with constant number of zeros or ones, ending with ones or zeros, etc.) but I can't find an example to demonstrate my claim that $T\neq\sigma(T^0,T^1)$.
Here are the relevant definitions.
Let $\Omega = \{0,1\}^{\mathbb N}$ the set of all binary sequences.
If $\mathcal F_n$ is a family of $\sigma$-algebras, we define $$T_n=\sigma\left(\bigcup_{k\geq n}\mathcal F_k\right)\qquad T^0_n=\sigma\left(\bigcup_{k\geq n}\mathcal F_{2k}\right)\qquad
T^1_n=\sigma\left(\bigcup_{k\geq n}\mathcal F_{2k-1}\right)$$
and
$T=\bigcap\limits_nT_n$ the tail $\sigma- $algebra,
$T^0=\bigcap\limits_nT^0_n$ the even tail $\sigma-$ algebra,
$T^1=\bigcap\limits_nT^1_n$ the odd tail $\sigma-$ algebra.
 A: My suggestion for a solution, based on @saz's comment:

We look at $\Omega=\{0,1\}^\mathbb N$ as the product of $\{0,1\}$ with the discrete topology $\mathbb N$ times.
For each $n$, we define $\pi_n:\Omega\rightarrow\{0,1\}$ the projection on the $n$-th coordinate, and $\mathcal F_n=\sigma(\pi_n)$, that is, $\mathcal F_n=\{\Omega,\varnothing,\pi_n^{-1}(0),\pi_n^{-1}(1)\}$.
We then define $T_n$, $T$, $T^0_n$, $T^1_n$, $T^1$ and $T^0$ as in the question.
Let $A$ denote the set of sequences $(x_n)$ such that $x_{2n}=x_{2n+1}$ infinitely often. 
Then $A$ is a tail event since omitting a finite number of indexes from such a sequence is not changing that $x_{2n}=x_{2n+1}$ occurs infinitely often. Thus, $T$ is not a trivial $\sigma$-algebra. 
Claim: $T^0$ is a trivial $\sigma$-algebra. 
Proof: for every $k>m$, $\mathcal F_{2k}\cap \mathcal F_{2m}=\mathcal F_{2k}$ hence, for every $n$, $\bigcap\limits_{1\leq k\leq n}\mathcal F_{2k}=\mathcal F_{2n} $ and the intersection of sigma-algebras is a sigma-algebra.
Similarly, $T^1$ is a trivial sigma-algebra and so is $\sigma(T^0,T^1)$.
Reminder, I have already shown that $\sigma(T^0,T^1)\subset T$.

Is this proof good enough to show that the inclusion $\sigma(T^0,T^1)\subset T$ can be strict?
