A jacket was originally priced $\textdollar 100$. The price was reduced by $10\%$ three times and increased by $10\%$ four times in some order. To the nearest cent, what was the final price?
1. **Initial Price Reduction**:
   The original price of the jacket is $\$100$. The price is reduced by $10\%$ three times. Each reduction can be represented as multiplying by $0.9$ (since $100\% - 10\% = 90\% = 0.9$).

   \[
   \text{Price after first reduction} = 100 \times 0.9 = 90
   \]

   \[
   \text{Price after second reduction} = 90 \times 0.9 = 81
   \]

   \[
   \text{Price after third reduction} = 81 \times 0.9 = 72.9
   \]

   Alternatively, we can combine these steps into one calculation:

   \[
   \text{Price after three reductions} = 100 \times (0.9)^3 = 100 \times 0.729 = 72.9
   \]

2. **Subsequent Price Increase**:
   The price is then increased by $10\%$ four times. Each increase can be represented as multiplying by $1.1$ (since $100\% + 10\% = 110\% = 1.1$).

   \[
   \text{Price after first increase} = 72.9 \times 1.1 = 80.19
   \]

   \[
   \text{Price after second increase} = 80.19 \times 1.1 = 88.209
   \]

   \[
   \text{Price after third increase} = 88.209 \times 1.1 = 97.0299
   \]

   \[
   \text{Price after fourth increase} = 97.0299 \times 1.1 = 106.73289
   \]

   Alternatively, we can combine these steps into one calculation:

   \[
   \text{Price after four increases} = 72.9 \times (1.1)^4 = 72.9 \times 1.4641 = 106.73289
   \]

3. **Rounding to the Nearest Cent**:
   The final price, rounded to the nearest cent, is:

   \[
   \boxed{106.73}
   \]