Given a circular necklace with $mn$ beads, each being either red or blue, we need to determine all the possible values of the ordered pair $(m, n)$ such that when the necklace is cut into $m$ blocks of $n$ consecutive beads, each block has a distinct number of red beads.

### Analysis

1. Understanding the Problem:
We have a necklace with $mn$ beads. We are tasked with cutting this necklace into $m$ segments, each containing $n$ consecutive beads. Each segment must have a distinct count of red beads. Since the necklace is circular, after the first $m-1$ segments are defined, the remaining beads automatically determine the last segment.

2. Constraints:
- Each of the $m$ segments must have a unique count of red beads.
- The total number of blocks is $m$, so the number of unique counts of red beads in these segments is also $m$.

3. Maximizing Unique Segment Counts:
- The possible range for the number of red beads in a segment is from 0 to $n$.
- For $m \leq n + 1$, it is possible to give each segment a unique number of red beads in this range.
- If $m > n + 1$, we cannot find enough distinct numbers of red bead distributions within segments since we only have $n + 1$ possible distinct numbers (0 through $n$).

### Conclusion

Let's summarize our findings:
- Each segment can have between 0 and $n$ red beads.
- To ensure each configuration is unique, $m$ cannot exceed $n + 1$.
- Hence, all possible ordered pairs $(m, n)$ that satisfy the conditions are such that $m \leq n + 1$.

Therefore, the solution is such that:
$$(m, n) \text{ is valid if } m \leq n + 1.$$

Thus, the final condition for the ordered pair is:
$$\boxed{(m, n) \text{ such that } m \leq n + 1}$$