A representation of the Clifford algebra could be obtained by creation and annihilation fermionic modes $c_k$ and $c_k^{\dagger}$, by the following definition:
$$
\Gamma_{2k-1}=c_k+c_k^{\dagger}\,\,;
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\Gamma_{2k}=i(c_k-c_k^{\dagger})
$$
For $k$ being an integer between $0 <k< (d+1)/2$. For directions $\mu$ with negative signature you should put a imaginary unit $i$ up front. The Lorentz generator of
$$
\delta [...]^{\rho}=\frac{1}{2}\theta_{\mu\nu}(\eta^{\rho\nu}[...]^{\mu})
$$
where $\theta_{\mu\nu}=-\theta_{\nu\mu}$ is given by:
$$
\Sigma(\theta)=-\frac{i}{2}\theta_{\mu\nu}\Gamma^{\mu}\Gamma^{\nu}
$$
Note that for "rotations" in $(2k,2k-1)$ plane is given by
$$
\Sigma(\theta)=-\frac{i}{2}\theta_{2k,2k-1}(c_k+c_k^{\dagger})(i)(c_k-c_k^{\dagger})=\frac{1}{2}\theta_{2k,2k-1}(-c_kc_k^{\dagger}+c_k^{\dagger}c_k)=
$$
$$
=\theta_{2k,2k-1}\left(c_k^{\dagger}c_k-\frac{1}{2}\right)=\theta_{2k,2k-1}\left(n_k-\frac{1}{2}\right)
$$
This provides the interpretation of the modes. The modes are the eigenvalues the $\Sigma^{2k,2k-1}$ generator, being $+1/2$ for $n_k=1$ and $-1/2$ for $n_k=0$. The same as $J_3$ for the $d=3$ case.
For even dimensions, there is a chiral matrix commutes with the Lorentz generators (e.g. $\gamma_5$ for $d=4$) and can be written in terms of the modes as:
$$
\bar{\Gamma}=\prod_k(1-2n_k)
$$
This means that the eigenvalues of $\bar{\Gamma}$, the chirality, is $+1$ if there is a even number of occupied modes and is $-1$ for odd. Since this matrix commutes with Lorentz generator, for even dimensions we may split a Dirac spinor into two spinors, a chiral spinor ($\bar\Gamma =+1$) and anti-chiral spinor ($\bar\Gamma =-1$).
For odd dimensions one of this gamma matrices will be left over. For the last $k$-mode we take just the $\Gamma_{2k-1}$ and the $\Gamma_{2k}$ is unused. You may note that now there are two matrices that commutes with Lorentz generator, the $\Gamma_{2k}$ and the $\bar\Gamma$. It is important to note that they not commute with each other, so we can use just one of them to split the representation but not both. Whatever matrix we use we will end up with a reduction of the number of modes by one. If there is $n$ modes for $d=2n$, for $d=2n-1$ we have $n-1$ modes.
There is also other way to obtain the odd dimension gamma matrices for $d=2n-1$ by starting with $d=2(n-1)$ and using the $\bar\Gamma$ matrix as the $\Gamma^{d}$. This have $n-1$ modes from the start.
Now you can see that there is a distinction between odd and even dimensions. So, is always expected to have this jumps on the numbers of components when you reduce an even dimension $d$ to $d-1$, or increase an odd dimension $d$ to $d+1$, because of the increasing number of $c_k$ modes.
