To address the problem, let's analyze the strategy needed for the magician to identify the polynomial $P(x)$ of degree $n$ based on the $2n$ values provided in non-decreasing order on the blackboard.

Given:
- The magician knows a positive integer $n$.
- The magician knows ordered real numbers $x_1 < x_2 < \dots < x_{2n}$.
- The magician is given the non-decreasing values $P(x_1), P(x_2), \dots, P(x_{2n})$ (but not which value corresponds to which $x_i$).

The polynomial $P(x)$ of degree $n$ has real coefficients and is determined by these $n+1$ coefficients, which we will refer to as $a_0, a_1, \ldots, a_n$.

Key Insight:
A polynomial $P(x)$ of degree $n$ with real coefficients can be expressed as:
$$P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0.$$

### Requirements and Limitations:

- The main challenge for the magician is that the specific pairing of the $2n$ values with the $x_i$'s is unknown, due to the reordering in non-decreasing sequence.
- This reordering could correspond to any permutation of the $2n$ original calculated values, masking the association with the specific $x_i$'s.
- Since a polynomial of degree $n$ can have at most $n$ distinct real roots, knowing the specific values doesn't directly provide the necessary associations to determine the coefficients, given the non-unique correspondence from $2n$ possible matches.

### Conclusion:

Considering the above analysis and constraints, the inability to uniquely determine $P(x)$ arises from the excess potential permutations and combinations inherent in the non-decreasing untagged order, making it impossible to exactly ascertain the coefficients of the polynomial.

Thus, the magician cannot uniquely determine the polynomial $P(x)$ with the given setup and constraints. The conclusion is that there is no strategy for the magician to perform the trick successfully.

Hence, the final answer is:
$$\boxed{\text{No}}$$