To solve this problem, we need to:

1. Formulate the six-digit number: Let the six-digit number formed by the cards be $\text{ABCDEF}$, where each letter represents a digit on the opposite side of each card.

2. Use divisibility by 11: According to the divisibility rule for 11, a number $\text{ABCDEF}$ is divisible by 11 if: $(A + C + E) - (B + D + F) \equiv 0 \pmod{11}$ or equivalently, if this difference is divisible by 11.

3. Determine constraints on digits: Since each card has a unique digit from 0 to 9 and all digits on the cards are unique, the six-digit number $\text{ABCDEF}$ will use six distinct digits.

4. Maximize and minimize: We aim to find both the smallest and largest possible six-digit numbers $\text{ABCDEF}$ that satisfy the divisibility rule.

Step 1: Finding the Smallest Possible Six-Digit Number

Let’s start by constructing the smallest number by using the smallest six digits (0 through 5) in increasing order and checking for the divisibility rule. If this set doesn't satisfy divisibility by 11, we’ll try other combinations while keeping the number as small as possible.

1. Smallest possible candidates: Start with 102345 (since the number can’t start with 0).
   • Calculate $(1 + 3 + 5) - (0 + 2 + 4) = 9 - 6 = 3$, which is not divisible by 11.
2. Continue checking upwards with the next smallest sets of digits until we find one that meets the criteria.

Step 2: Finding the Largest Possible Six-Digit Number

For the largest possible number, we start with the largest six digits (9 through 4 in decreasing order) and check each possibility in a similar way.