Given a pair $L_1,L_2$ of latin squares of order $n$, let $r=N(L_1,L_2)$ denote the number of distinct ordered pairs which occur when $L_1$ and $L_2$ are superimposed. Consider now a set of $t$ latin squares of order $n$, and denote by $r_n(t)$ the number of distinct pairs which arise when each possible pair of distinct squares is checked to determine their level of $r$-orthogonality. More details and examples can be found here.
The spectrum for latin squares of order $n$ is the set of all positive values that can occur as $r_n(t)$ orthogonalities.
The spectrum of $r$-orthogonalities of sets of $2$ latin squares is given by the following theorem.
The spectrums of $r_n(t)$-orthogonalities for $n \leq 6$ are the following, note that when we say $x - y$ in a spectrum we mean that $x$, $y$ and every value in between are included in the spectrum:
The following tables show how many sets of $t$ latin squares of order $n$ produce each of the possible $r_n(t)$-orthogonalities.
\[ \begin{array}{lcr}
\textbf{n=3} \mbox{ and } \textbf{t=2} \\
r_3(2) & \mbox{number of sets} \\
3 & 1 \\
9 & 2 \end{array}\]
\[ \begin{array}{lcr}
\textbf{n=4} \mbox{ and } \textbf{t=2} \\
r_4(2) & \mbox{number of sets} \\
4 & 4 \\
6 & 12 \\
8 & 6 \\
9 & 24 \\
12 & 48 \\
16 & 2\end{array}\]
\[ \begin{array}{lcr}
\textbf{n=4} \mbox{ and } \textbf{t=3} \\
r_4(3) & \mbox{number of sets} \\
12 & 4 \\
16 & 36 \\
20 & 54 \\
21 & 72 \\
22 & 72 \\
26 & 144 \\
27 & 192 \\
28 & 144 \\
30 & 576 \\
32 & 198 \\
33 & 504 \\
34 & 72 \\
36 & 198 \\
40 & 36 \\
48 & 2 \end{array}\]
\[ \begin{array}{lcr}
\textbf{n=5} \mbox{ and } \textbf{t=2} \\
r_5(2) & \mbox{number of sets} \\
5 & 56 \\
7 & 200 \\
10 & 600 \\
11 & 1200 \\
12 & 1200 \\
13 & 6150 \\
14 & 5400 \\
15 & 14640 \\
16 & 14400 \\
17 & 20050 \\
18 & 6400 \\
19 & 4800 \\
21 & 150 \\
25 & 18 \end{array}\]
\[ \begin{array}{lcr}
\textbf{n=5} \mbox{ and } \textbf{t=3} \\
r_5(3) & \mbox{number of sets} \\
15 & 56 \\
19 & 600 \\
24 & 1800 \\
25 & 1800 \\
27 & 5400 \\
29 & 12600 \\
30 & 1200 \\
31 & 36450 \\
32 & 12600 \\
33 & 42600 \\
34 & 72000 \\
35 & 137520 \\
36 & 141000 \\
37 & 261000 \\
38 & 486000 \\
39 & 721050 \\
40 & 894600 \\
41 & 1790400 \\
42 & 2656800 \\
43 & 3904500 \\
44 & 5434200 \\
45 & 8391640 \\
46 & 9432000 \\
47 & 12768750 \\
48 & 12945000 \\
49 & 13424400 \\
50 & 9972000 \\
51 & 8275500 \\
52 & 4489200 \\
53 & 2820600 \\
54 & 1156800 \\
55 & 582714 \\
56 & 144000 \\
57 & 97200 \\
58 & 7200 \\
59 & 27900 \\
63 & 5700 \\
75 & 36 \end{array}\]