Is there a companion algorithm for OTP to ensure integrity and/or authentication?
You can construct a one-time MAC that has similar properties to the OTP. Better still, it uses a fixed number of bits for each message.

Here's how it works. Choose the closest prime to your message block size. Let's say you plan to process 128-bit chunks of your message. Let's say there are $L$ such blocks.

The first job is to pick the first prime larger than $2^{128}$ which is $2^{128}+51$. Call this value $q$.

The key has two pieces. Let's call these $x$ and $y$ and choose these at random from the range $1$ to $q-1$.

Your MAC is then: 

$mac(m, x, y) = m_L\times x^L + m_{L-1} \times x^{L-1} + ... + m_{1} \times x + y \mod q $

You basically evaluate a polynomial with a term for each block of the message. The co-efficient is the message block and this is multiplied by the key raised to the power of the current block number. All operations are done mod $q$.

Provided $x$ and $y$ and are used precisely once, this MAC is secure against all adversaries.