To determine all positive integers $n \geq 2$ that satisfy the given condition, we need to analyze when $a \equiv b \pmod{n}$ if and only if $ab \equiv 1 \pmod{n}$ for all $a$ and $b$ that are relatively prime to $n$.

### Step 1: Analyze the given condition

The problem requires:
- $a \equiv b \pmod{n}$ if and only if $ab \equiv 1 \pmod{n}$.

### Step 2: Translating the conditions

1. If Part: If $a \equiv b \pmod{n}$, then $a = b + kn$ for some integer $k$. So, $ab \equiv 1 \pmod{n}$ implies $(b + kn)b \equiv 1 \pmod{n}$.

2. Only If Part: If $ab \equiv 1 \pmod{n}$, then there exists some integer $x$ such that $ab = 1 + xn$. This situation implies $a \equiv b \pmod{n}$.

Consider using group theory concepts, including units modulo $n$. The set of integers coprime with $n$, under multiplication modulo $n$, forms the **multiplicative group of units mod $n$**, denoted by $\mathbb{Z}_n^*$.

### Step 3: Conditions on the structure of $\mathbb{Z}_n^*$

For both conditions to hold:
- $\mathbb{Z}_n^*$ forms a group where every element has its inverse to satisfy the divisors such that $(ab)^2 \equiv 1 \pmod{n}$.
- Specifically, $a^2 \equiv 1 \pmod{n}$ for all $a$ in $\mathbb{Z}_n^*$, indicating that each element in $\mathbb{Z}_n^*$ is its own inverse.

### Step 4: Determine $n$ such that every unit in $\mathbb{Z}_n^*$ is its own inverse

To solve the problem, every element in $\mathbb{Z}_n^*$ must be its own inverse. This is equivalent to demanding that the group order must be a power of 2, as groups with elements all self-invertible are those isomorphic to elementary abelian 2-groups.

### Step 5: Identifying all eligible $n$

From the conditions operating on $\mathbb{Z}_n^*$, the integer $n$ can be characterized as the product of distinct prime powers where the $\phi(n) = 2^k$ for some $k$, and where each $\phi(p_i^{k_i})$, remains power of two.

For $n \ge 2$, this includes $n = 2^k (k \geq 1), 2^k3 (k \geq 0)$, resulting into:
- $n = 2, 3, 4, 6, 8, 12, 24$.

Thus, the list of positive integers $n \geq 2$ satisfying the conditions of the problem statement are:
$$\boxed{2, 3, 4, 6, 8, 12, 24}.$$

This completes the problem's solving process.