Determine all monic polynomials $ p(x)$ of fifth degree having real coefficients and the following property: Whenever $ a$ is a (real or complex) root of $ p(x)$, then so are $ \frac{1}{a}$ and $ 1\minus{}a$.
Given a monic polynomial \( p(x) \) of fifth degree with real coefficients, we need to determine all such polynomials that satisfy the property: if \( a \) is a root of \( p(x) \), then \( \frac{1}{a} \) and \( 1 - a \) are also roots.

1. **Identify the roots and their properties:**
   - If \( a \) is a root, then \( \frac{1}{a} \) and \( 1 - a \) are also roots.
   - Consider the set of roots: \( \{a, \frac{1}{a}, 1 - a, \frac{a - 1}{a}, \frac{1}{1 - a}, \frac{a}{a - 1}\} \).

2. **Analyze the possible values of \( a \):**
   - Since the polynomial has real coefficients, the roots must come in conjugate pairs if they are complex.
   - We need to find values of \( a \) such that the set of roots is consistent with the given property.

3. **Case 1: \( a = \frac{1}{a} \):**
   - This implies \( a^2 = 1 \), so \( a = \pm 1 \).
   - If \( a = -1 \), the set of roots is \( \{-1, -1, 2, 2, \frac{1}{2}, \frac{1}{2}\} \).

4. **Case 2: \( a = 1 - a \):**
   - This implies \( 2a = 1 \), so \( a = \frac{1}{2} \).
   - The set of roots is the same as in Case 1: \( \{-1, -1, 2, 2, \frac{1}{2}, \frac{1}{2}\} \).

5. **Case 3: \( a = \frac{a - 1}{a} \):**
   - This implies \( a^2 = a - 1 \), so \( a^2 - a + 1 = 0 \).
   - The roots are \( a = \frac{1 \pm i\sqrt{3}}{2} \).
   - The set of roots is \( \left\{\frac{1 - i\sqrt{3}}{2}, \frac{1 + i\sqrt{3}}{2}, \frac{1 + i\sqrt{3}}{2}, \frac{1 - i\sqrt{3}}{2}, \frac{1 - i\sqrt{3}}{2}, \frac{1 + i\sqrt{3}}{2}\right\} \).

6. **Case 4: \( a = \frac{1}{1 - a} \):**
   - This is equivalent to Case 3.

7. **Case 5: \( a = \frac{a}{a - 1} \):**
   - This is equivalent to Case 1.

8. **Construct the polynomial:**
   - For real coefficients, we cannot have a polynomial with roots only from the complex set in Case 3.
   - Therefore, the polynomial must have roots from the real set in Cases 1 and 2.

9. **Form the polynomial:**
   - The roots are \( \{-1, 2, \frac{1}{2}\} \).
   - The polynomial can be written as \( P(x) = (x + 1)^p (x - 2)^q (x - \frac{1}{2})^r \) with \( p, q, r > 0 \) and \( p + q + r = 5 \).

10. **Include the complex roots:**
    - The polynomial can also include the complex roots from Case 3, which form the quadratic factor \( x^2 - x + 1 \).

11. **Final polynomial:**
    - The polynomial can be written as \( P(x) = (x + 1)(x - 2)(x - \frac{1}{2})(x^2 - x + 1) \).

The final answer is \( \boxed{ P(x) = (x + 1)(x - 2)(x - \frac{1}{2})(x^2 - x + 1) } \).