The answer would depend on a lot of factors, such as what is the government money spent on, what is the tax regime etc. For starters, let us look at what would happen in different time scales. We can use a simple textbook IS-LM-AS model with three equations:

$$Y = C(Y-T) + I(r) + G$$ $$M^s(H, P) = M^d(Y, r)$$ $$Y = S(P)$$

The first equation is the IS equation. The total income in the economy (LHS) is equal to the total expenditure, which is the sum of consumption $C$, investment $I$ and government spending $G$. $C$ rises with disposable income $Y-T$ where $T$ is tax, but not as much as income as people save part of it. Therefore $0 < \partial Y/ \partial (Y-T) < 1$. $I$ falls with the interest rate $r$, as a higher interest rate means more difficulty getting funds. In the LM equation, real money supply $M^s$ increases with hard money $H$ as there is more money in people's hands and falls with price level $P$ as each note buys less when price is high. Money demand $M^d$ rises with $Y$ as higher income households would need more money to accommodate higher spending, assuming constant velocity. $M^d$ falls with $r$ as bond prices are inversely related to $r$, so with high $r$ and low pond prices, people would hold bonds instead of money. Finally, the AS curve states that total supply $S$ is related to $P$. The result will depend on the nature of this relation.

We have $Y$, $r$ and $P$ as endogenous variables. Rest are all exogenous. To avoid notational clutter, I will write $A_B$ for the derivative of $A$ with respect to $B$. For example, $Y_G$ is the derivative of $Y$ with respect to $G$. Differentiating both sides of all the three equations and solving for $Y_G$, we get,

$$Y_G = \frac{M^d_r S_p}{S_P (1-C_{Y-T})M^d_r + S_p M^d_Y I_r - M^s_P I_r}$$

A similar exercise for $Y_T$ yields

$$Y_T = \frac{- C_{Y-T} M^d_r S_p}{S_P (1-C_{Y-T})M^d_r + S_p M^d_Y I_r - M^s_P I_r}$$

Suppose the government spending has changed by $\Delta G$. The $Y$ changes approximately by $Y_G \Delta G$. Suppose $T$ has changed by $\Delta T$, then $Y$ changes by approximately $Y_T \Delta T$. In your question, the government has reduced spending exactly as much as it has reduced taxes. Therefore, $\Delta T = \Delta G$. Using this, the total change in $Y$ is approximately

$$(Y_G + Y_T) \Delta G$$

$$ = \frac{(1 - C_{Y-T}) M^d_r S_p}{S_P (1-C_{Y-T})M^d_r + S_p M^d_Y I_r - M^s_P I_r} \Delta G$$

The interesting thing to note is, $Y_T = - C_{Y-T} Y_G$. Why is it so? Looking at the processes helps. When the government spends an additional penny, that spending directly gets added to $Y$ in the first step. Once that penny reaches the people, they spend $C_{Y-T}$ of it, which also gets added to $Y$. But this additional $C_{Y-T}$ further prompts additional spending and so on. But in the case of raising $T$, the raise in T does not directly count as a fall in $Y$. Raising $T$ leads to falling expenditure, which in the first round only amounts to a fall of $C_{Y-T}$ in $Y$. This fall reduces expenditure further and so on. Therefore, the first round effect of changing $T$ is $C_{Y-T}$ times of the first round effect of changing $G$ and the rest of the process is same.

As $0 < C_{Y-T} < 1$, this means that changing $T$ and $G$ by the same amount would lead to the two having effects opposite in direction, but the effect of $G$ would be higher. Overall, the direction would be the same as a change in $G$ only.

Reducing $G$ and raising $T$ would have the same direction of effect as reducing $G$ and doing nothing to $T$.

The LM and AS curves add more nuance but the result remains more or less the same. With price stickiness, in the short run the price level is a constant. Therefore, the $AS$ curve is horizontal and $S_P = \infty$. In the equation above, as we do $ \lim_{P \rightarrow \infty} $, we get

$$(Y_G + Y_T) \Delta G = \frac{(1 - C_{Y-T}) M^d_r}{(1-C_{Y-T})M^d_r + M^d_Y I_r} \Delta G$$

You can check that this is an overall negative quantity. You can check that if $I_r = 0$, $\Delta Y$ would have been exactly equal to $\Delta G$. But with $I_r < 0$, additional government spending eats up part of investment by raising interest rates, dampening the effect.

As $S_P$ reduces in the medium run, $S_P$ still remains positive. Oversimplifying, this is because higher $P$ means lower real wage, higher employment and therefore higher supply. With a positive $S_P$, it is easy to check that $(Y_G + Y_T) \Delta G$ is less fall in income than the short run, but still negative.

In the long run, $S_P$ becomes zero, and the income becomes independent of price level. Here, $(Y_G + Y_T) \Delta G = 0$, which was your hunch.

Therefore, refining the previous statement:

Reducing $G$ and raising $T$ would have the same direction of effect as reducing $G$ and doing nothing to $T$:

1. Reducing the income in the short run
2. Reducing the income but a little less in the medium run
3. Having no effect in the long run

The above simplifies a lot of things however. We had a lump sum tax. Different tax regimes would give different results, but as long as higher tax reduces disposable income, the direction of the results would remain the same. We have considered government spending affecting consumption directly only. Government expenditure in the form of investment subsidies might have different effect. Finally, the nature of expenditure cut also would affect the results. In our model, the long run AS income is exogenous: but it is very likely that it is not exogenous in the real world. Reduction in government spending on key sectors like infrastructure might negatively affect it.