Let $x_1, \dotsc, x_n \in \mathbb{R}^d$ and $\theta_1, \dotsc, \theta_n \in \mathbb{R}^d$ be vectors such that for every $k \in [n]$, the following inequality holds:
$$ \langle x_k, \theta_k \rangle < \langle x_1, \theta_k \rangle < \dotsb < \langle x_{k-1}, \theta_k \rangle < \langle x_{k+1}, \theta_k \rangle < \dotsb < \langle x_n, \theta_k \rangle, $$
where $\langle \cdot, \cdot \rangle$ denotes the standard inner product in $\mathbb{R}^d$.
Can it be shown that $n \leq d+1$? (or $O(d)$?)
For $d=2$ the maximal $n$ is $3=d+1$. Indeed, between 4 points $x_1,x_2,x_3,x_4$ either one of them, $x_k$, belongs to the convex hull of three others, then $\langle \theta,x_k\rangle$ can not be the minimal number between $\langle \theta,x_i\rangle$; or the segments $[x_1,x_j]$ and $[x_p,x_q]$ (where $\{1,j,p,q\}=\{1,2,3,4\}$) intersect, then it is impossible that $\langle \theta,x_1\rangle$ and $\langle \theta,x_j\rangle$ are both less then $\langle \theta,x_p\rangle$ and $\langle \theta,x_q\rangle$.
But already for $d=3$ there is no bound for $n$. Consider the points $x_k$ on the moment curve $(t, t^2,t^3)$ with parameters $t_k=3^k$. Then the polynomials $p_k(t)=t(t-t_k)^2$ satisfy $$p_k(t_k)<p_k(t_1)<p_k(t_2)<\dots <p_k(t_{k-1})<p_k(t_{k+1})<\dots$$ for every $k$. Equivalently, for the vectors $\theta_k=(t_k^2,-2t_k, 1)$ we have $$\langle x_k, \theta_k \rangle < \langle x_1, \theta_k \rangle<\langle x_2, \theta_k \rangle<\dots<\langle x_{k-1}, \theta_k \rangle <\langle x_{k+1} , \theta_k \rangle<\dots$$
This is not a complete answer, but too large to fit reasonably as a comment. As shown by Fedor Petrov, it is not true that $n \leq d+1$, since for $d = 3$ we can construct $5$ such points and directions. I will give another example for $d = 4$. Consider the $7$ points $x_1, \ldots, x_7 \in \mathbb{R}^4$ given by
$$ \mathbf{X} = \begin{bmatrix} 1 & 2 & 3 & 5 & 6 & 11 & 12\\ 2 & 1 & 100 & 101 & 102 & 103 & 104 \\ 2 & 3 & 4 & 1 & 100 & 101 & 102 \\ 2 & 3 & 4 & 5 & 6 & 1 & 100 \end{bmatrix}, $$
and the $7$ directions $\theta_1, \ldots, \theta_7 \in (\mathbb{R}^4)^*$ given by $$ \boldsymbol{\Theta} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 9800 & -249 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 9800 & 0 & -549 & 0 \\ 0 & 0 & 0 & 1 \\ 9800 & 0 & 0 & -1149 \end{bmatrix}. $$ The projections onto each direction are therefore given by $$ \boldsymbol{\Theta} \mathbf{X} = \begin{bmatrix} 1 & 2 & 3 & 5 & 6 & 11 & 12 \\ 2 & 1 & 100 & 101 & 102 & 103 & 104 \\ 9302 & 19351 & 4500 & 23851 & 33402 & 82153 & 91704 \\ 2 & 3 & 4 & 1 & 100 & 101 & 102 \\ 8702 & 17953 & 27204 & 48451 & 3900 & 52351 & 61602 \\ 2 & 3 & 4 & 5 & 6 & 1 & 100 \\ 7502 & 16153 & 24804 & 43255 & 51906 & 106651 & 2700 \end{bmatrix}, $$ which enjoy the stated property. Due to the way in which this example was constructed, I suspect $n = 2d - 1$ is attainable in general. Even if this is true, it does not provide an upper bound for $n$, but it may provide a starting point in our search for one.
